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www.centerforspace.comPg 1 of 120
Fundamentals of Astrodynamics and Applications
By David A. ValladoTutorial Lectures at the
4th ICATT, Madrid, SpainApril 30, 2010May 3-6, 2010
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Objectives
• Use an example problem to illustrate various astrodynamic techniques you’ll need to know
• Introduce you to the various topics that the text covers in more detail
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Problem Scenario
• Determine when you can see a satellite from a ground site
• What we’ll need to understand– Time– Coordinate systems– Propagation– Orbit Determination– ... and some others ☺
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What we’ll cover• Fundamental Concepts
– Time and Coordinate Systems• Newton
– Equations of Motion• Kepler
– Equation – Problem– Satellite state
• Perturbations/Propagation – Special– General
• Orbit Determination and Estimation• Applications
www.centerforspace.comPg 5 of 120
Fundamentals of Astrodynamics and Applications Third Edition
Space Technology Library (Vol 21), Microcosm Press/Springer
By David A. Vallado
Center for Space Standards and innovation
Paperback (ISBN 978-1-881883-14-2)
Published Spring 2007
http://astrobooks.com/index.asp?PageAction=VIEWPROD&ProdID=1137
US$ 60.00
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My Objective with the Book• Cover
– Fundamentals– Some advanced material
• Bridge the gap in between• Details
– Consistent notation
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• Fundamental Concepts• Newton• Kepler• Perturbations• Orbit Determination• Applications
Chapter 3
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Time and Coordinate Systems
• Essential, but not terribly exciting
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What Time is it?
• 14:28– Ok – That specifies that it’s afternoon– But what time zone?
• Mountain Time is 6/7 hours before UTC (Greenwich, Zulu)
– Need to specify » Daylight Savings» Standard Time
– Is that all? … No!• TAI, TT (TDT), TDB, TCB, TCG, GPS, …
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Solar and Sidereal TimeEarth Sun
Stars
Reference Direction
Solar day(24h)
Sidereal day(23h 56m 4.0905s)
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Greenwich and Local TimesLocal Meridian
StarLHAstar
VGMST
0°
GHA~
VLSTl
~
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Hour Angles vs Time?
• 24 hrs = 360 degrees– Sidereal time assumed
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Hour AnglesLocal Observer
Star
EastLHAstar
VGMST
GHAstar
VLST
l
~
0°
astar
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What Time is it? (continued)
• Additional times– UT1 (Universal Time, sidereal time)
• Solution from observations• Shows slowly decreasing Earth rotation rate
– UTC is Coordinated Universal Time (solar time)• “Clock time”• Maintained within 0.9 s of UT1
– Leap Seconds
• UTC = UT1 + ΔUT1– ΔUT1
» EOP Parameter that accounts for actual Earth rotation» Calculated by USNO/ IERS
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Time Scales
-40.0
-30.0
-20.0
-10.0
0.0
10.0
20.0
30.0
40.0
50.0
Jan-61 Jan-65 Jan-69 Jan-73 Jan-77 Jan-81 Jan-85 Jan-89 Jan-93 Jan-97 Jan-01 Jan-05 Jan-09 Jan-13
Diff
eren
ce in
Tim
e to
TA
I (s
ec)
UTC
UT1
TCB
TAI
GPS
TDBTCG
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Summary for time
• Time– Can be off by up to a second if no ΔUT1– TT can be off a minute
• Used for many calculations – Impact
• Seems small but …– Consider satellite traveling at 7 km/s
– Many conversions necessary• Satellite moves wrt sidereal time• Clocks record Solar time
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Coordinate Systems
• Sun based– Heliocentric– Barycentric
• Earth Based– Geocentric (Inertial and fixed)– Topocentric (fixed)
• Satellite Orbit Based– Perficoal– Radial vs Normal– Equinoctial
• Satellite Based– Attitude
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Heliocentric Coordinate System
Sun
Ecliptic Plane
Aphelion ~ 1 Jul
Perihelion ~ 1 Jan
Summer solstice1st day of Summer
~ Jun 21
Vernal equinox1st day of spring
~ Mar 21
Vernal Equinox1st day of winter
~ Dec 21
Autumnal Equinox1st day of Fall
~ Sep 23
Y
Z
X, ~^
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Geocentric and Ecliptic Coordinates
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Geocentric Coordinate System
I, ~^ J
K
Equatorial Plane
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Local Coordinate System
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Orbit Based Systems - Perifocal
I J
P
Q
W K
Perigee, closest point to Earth
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Orbit Based Systems – Normal and Radial
IJ
R, radial^
S, along-track^W, cross-track^
K
N
W
v, T, in-track^
v
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Orbit Based Systems - Equinoctial
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Angular Measurements
• Latitude and longitude– Familiar
• Right Ascension-Declination– Optical measurements
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Right Ascension - Declination
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Motion of the Coordinate System
• Earth’s orbit is not exactly stable– Precession
• Long period movement (~26000 years)– Nutation
• Short period movement (~18.6 years)
• Fixed vs Inertial– Sidereal Time
• Polar Motion– Axis of rotation moves slightly over time
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Precession and NutationLuni-solar precession effect Nutation effect
Ecliptic planePlanetary effect
Earth’s orbitAbout Sun
Precession of Equinox
Earth’s equator
~
e
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Polar Motion
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Celestial Reference Frame
Terrestrial Reference Frame
IAU 1976 (zA ,θA, ζA)
∆ε, ∆ψ(Tables)
1984 ProceduresTraditional Traditional
Interpolation
R3[θGAST · 1982]
IAU 1982 (∆ε, ∆ψ , ε0 , δ∆ε, δ ∆ψ )
R2[-xp] R1[-yp]
[PN]
Sidereal Rotation
Polar Motion
PrecessionNutation
MOD
TOD
PEF
ITRF
Equinox based
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Celestial Reference Frame
Terrestrial Reference Frame
MHB2000 (zA ,θA, ζA) MHB2000 (χA, ωA, ψA, ε0) X, Y(Series)
X, Y, s(Tables)
R3[θERA ]
2003 Procedures
X, Y[BPN]
s (Series)
Traditional Canonical 4-term Rotation
Series TraditionalInterpolation
NonRotating Origin
R3[θGAST · 2000 ]
MHB2000 (∆ε, ∆ψ , ε0)
R1[-yp] R2[-xp]
[BPN]
Sidereal Rotation
Polar Motion
Bias-PrecessionNutation
R1[-yp] R2[-xp] R3[s’ ]
Eq. (xxx) = f ( X, Y, s )
CIRS
TIRS
ITRF
MOD
ERS
TIRS
ITRF
Equinox based CIO based
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Terrestrial Reference Frame
P03 (zA ,θA, ζA) P03 (χA, ωA, ψA, ε0) X, Y(Series)
X, Y, s(Tables)
R3[θERA ]
2006 Procedures
X, Y[BPN]
Traditional Canonical 4-term Rotation
Series TraditionalInterpolation
NonRotating Origin
R3[θGAST · 2006 ]
MHB2000 (∆ε, ∆ψ , ε0) + optional 2006 rate adjustments
R1[-yp] R2[-xp]
[BPN]
Sidereal Rotation
Polar Motion
Bias-PrecessionNutation
R1[-yp] R2[-xp] R3[s’ ]
Eq. (xxx) = f ( X, Y, s )
P03 (εA, ψJ, φJ, γJ)
Fukushima –Williams
s (Series) or EO
CIRS
TIRS
ITRF
MOD
ERS
TIRS
ITRF
Equinox based CIO based
Celestial Reference Frame
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Earth’s shape
• Oblate Spheroid– An ellipsoidal approximation
• Other terms– Geoids
• Gravity acts equally at all points on this surface– Plumb-bobs will hang perpendicular
– Geopotential• Mathematical representation of the precise
gravitational effect
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Earth Surface
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Earth Ellipsoid
• Convert geocentric (φgc) and geodetic (φgd) latitude
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• Fundamental Concepts• Newton• Kepler• Perturbations• Orbit Determination• Applications
Chapter 1
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Newton’s Laws• 1. Every body continues in its state of rest, or of uniform motion in a
right [straight] line, unless it is compelled to change that state by forces impressed upon it.
• 2. The change of motion is proportional to the motive force impressed and is made in the direction of the right line in which that force is impressed.
• 3. To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal and directed to contrary parts. (Newton [1687] 1962, 13)– The third law in Newton’s own words:
• If a horse draws a stone tied to a rope, the horse (if I may say so) will be equally drawn back towards the stone; for the distended rope, by the same endeavor to relax or unbend itself, will draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. (Newton [1687] 1962, 14)
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Gravitational Law
• Forms the basis of Two-body dynamics– G is constant of gravitation =
6.673x10-20 km3/kgs2rr
rmGm
f satgravity 2
⊕−=
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Two-body Equation of Motion
• Simple form resulting from
rr
rmmG
r sat2
)( +−= ⊕
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• Fundamental Concepts• Newton• Kepler• Perturbations• Orbit Determination• Applications
Chapter 2
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Kepler’s Laws
• 1. The orbit of each planet is an ellipse with the Sun at one focus.
• 2. The line joining the planet to the Sun sweeps out equal areas in equal times.
• 3. The square of the period of a planet is proportional to the cube of its mean distance to the Sun.
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Conic Sections
• All orbits follow– Circle– Ellipse– Parabola– Hyperbola– Rectilinear
Ellipse
Circle
PointParabola
Hyperbola
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Nomenclature
• Kepler’s Equation and Kepler’s Problem– Very different!– Kepler’s equation
• Found during Kepler’s analysis of the orbit of Mars– Kepler’s problem
• Generically used for propagating a satellite forward– Usually two-body dynamics
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Kepler’s Equation
• Find Eccentric anomaly (E)– E = 0º at ν = 0º,
180º
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Kepler’s Problem
• Find future position and velocity– Given starting state– Called propagation
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Satellite State Representations• Convey location of a satellite in space and time• Types
– Numerical• Position and velocity vectors
– Analytical (Elements)• Classical (Keplerian, Osculating, two-body) (a, e, i, Ω, ω, ν) • Equinoctial (af, ag, L, n, χ, ψ)• Flight (λ, φgc, φfpa, β, r, v) • Spherical (α, δ, φfpa, β, r, v) • Canonical
– Delaunay– Poincare
• Mean elements (theory dependant)– Two-line element sets
» AFSPC, SGP4 derived, ‘mean’ elements– ASAP– LOP– Other
• Other– Semianalytical
• Theory dependant
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Classical Orbital Elements
Line of nodes, n
Angular momentum, h
Perigee, e
I
r
v
i υ
ω
Ω
Equatorial Plane
J
K
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• Fundamental Concepts• Newton• Kepler• Perturbations• Orbit Determination• Applications
Chapter 8/9
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Introduction
• Several forces affect satellite orbits– Gravitational– Atmospheric Drag– Third Body
• Sun, Moon, planets– Solar Radiation Pressure– Tides
• Solid Earth, Ocean, pole, etc.– Albedo– Thrusting– Other
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Applicability
Orbital Altitude (km)
Central Body Gravity
Solid Earth Tides
Ocean Tides
Albedo
Solar Radiation Pressure
Atmospheric Drag
Other
Third Body Gravity
100 1000 10,000
100,000
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Perturbations
Time
Mean Change
Mean Change
Short-periodic plus long-periodic, and secular
Secular
Long-periodic andsecular
t1 t2t3 t4
c
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Central Body Gravitational Forces
• Largest single contributor to the motion– It’s why satellites stay in orbit!
• Conservative force– Total kinetic and potential energy remains the same
.0for 2 and ,0for 1with
,21
)!()12(
)!( ,
21
)!(
)!()12(
≠===
−+
+=
+
−+=
⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡
mkmk
mnkn
mnandP
mn
mnknP
nm
nm
nm
nm
SC
SC
nmnm
( ) ( ) ( ) = ( )
P dd
Pnm nm
m
msin cossin
sinφ φφ
φ
( ) ( ) = !
(
- P ddn n
m
m
n
nsin
sin )sinφ
φφ1
212
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡+⎟
⎠⎞
⎜⎝⎛
∑∑==
∞
=
)sin cos)((sin + 1 EE02
λλφ mnmSmnmCnmPn
ra
rGMV
n
mn
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Zonal Harmonics
Top
Side
2, 0 3, 0 4, 0
5, 0
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Sectoral Harmonics
Top
Side
l = 2 l = 3 l = 4
l = 5
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Tesseral Harmonics
Top
Side
2, 1 3, 1 3, 2 4, 1
6, 4
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Nodal Regression
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Nodal Regression
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Apsidal Rotation
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Apsidal Rotation
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Gravitational Effects
• Long ago when computers were slow…• Gravitational modeling
– Often square gravity field truncations• Appears the zonals contribute more
– Point to take away:• Use “complete” field• Any truncations should include additional, if not all,
zonal harmonics
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Gravitational Modeling
• Satellite JERS (21867)– Comparison to 12x12 field– Note the variability over time
• 22x22 vs 18x18 and 70x22 vs 70x18
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
400.0
0.0 1440.0 2880.0 4320.0 5760.0
Time, min from Epoch
Diff
eren
ce (m
)
22x22
20x20
18x18
16x16
14x14
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
400.0
0.0 1440.0 2880.0 4320.0 5760.0
Time, min from Epoch
Diff
eren
ce (m
)
70x22
70x20
70x18
70x16
70x14
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Atmospheric Drag
• Large force for near-Earth satellites– Very difficult to model
• Non-conservative force– Total kinetic and potential energy not constant
• Heat, other losses through friction
rel
relrel
Ddrag v
vvm
Aca 2
21 ρ−=
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Drag Effect on Orbits
Orbit tends to circularize
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Available Data
0.0
50.0
100.0
150.0
200.0
250.0
300.0
Jan-50 Jan-54 Jan-58 Jan-62 Jan-66 Jan-70 Jan-74 Jan-78 Jan-82 Jan-86 Jan-90 Jan-94 Jan-98 Jan-02 Jan-06 Jan-10
Solar Cycle 23
Solar Cycle 22
Solar Cycle 21
Solar Cycle 20
Solar Cycle 19
F 10.7
ctr F 10.7
avg a p
Trend
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Space Weather – Predictions
• Lots of Variability– Constant F10.7
• Not very accurate• Never use 0.0!
– Schatten• Varies with each solar cycle
– Polynomial Trend – Matches several solar cycles
• F10.7 = 145 + 75*COS{ 0.001696 t + 0.35*SIN(0.001696 t )} – t is the number of days from Jan 1, 1981
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Observed vs Adjusted Solar Flux
-40.0
-30.0
-20.0
-10.0
0.0
10.0
20.0
30.0
40.0
Jan-50 Jan-54 Jan-58 Jan-62 Jan-66 Jan-70 Jan-74 Jan-78 Jan-82 Jan-86 Jan-90 Jan-94 Jan-98 Jan-02 Jan-06
DRAO (obs) - Lenhart (adj)
data
DRAO (obs) - DRAO (adj)
data
DRAO (adj) - Lenhart (adj)
data
• Data errors– Some
inconsistencies• 10-40 SFU
– Which does the model require?
• MSIS– Observed
• Others– Adjusted
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Solar Flux Predictions – Long Term
50
100
150
200
250
27-Nov-93 20-May-99 09-Nov-04 02-May-10 23-Oct-15 14-Apr-21 05-Oct-26
Trend
Schatten Nov 05
Schatten Oct 96
Schatten Sep 97
Last F10.7
Schatten Jul 02
Mon Avg
Schatten Sep 00
Schatten Jul 03
Schatten Mar 08
• Data differences– One solar cycle
• ~150 SFU
– Almost equal to the solar min-max difference!
Now
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Solar Flux Predictions – Shorter Term
• Data differences– Min, Mid, and
Max• 30-50 SFU
– Note timing of Cycle is off
0.0
50.0
100.0
150.0
200.0
250.0
300.0
Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04 Jan-06 Jan-08 Jan-10 Jan-12
Solar Cycle 23, May 1996 - April 2000 - March 2008
Jan 94
Jul 04
Trend
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Solar Flux Predictions – Shorter Term
• Early, Mid, and Late– Also 30-50 SFU
differences
0.0
50.0
100.0
150.0
200.0
250.0
300.0
Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04 Jan-06 Jan-08 Jan-10
Solar Cycle 23, May 1996 - April 2000 - March 2008
Apr 95Early
Apr 95Late
Trend
Apr 95Mid
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Solar Flux Predictions – Short Term
• NOAA Predictions– 27-day and 45-day (F10.7 and ap)– 3-day
• 3-hourly Kp values off significantly as well
-60
-50
-40
-30
-20
-10
0
10
20
30
40
0 7 14 21 28 35 42 49
Prediction time (days)
Sola
r Fl
ux D
iffer
ence
(SFU
)
27-day F 10.7
45-day F 10.7
-10
-5
0
5
10
15
20
0 7 14 21 28 35 42 49
Prediction time (days)
Geo
mag
netic
ap
Diff
eren
ce
27-day a p
45-day a p
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Simulated Sensitivity Analysis
• JERS sample orbit– Different
atmospheric models• Baseline
– Numerical propagation
– Jacchia-Roberts – 3-hourly
geomagnetic
– Relative comparison only
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
0 1440 2880 4320 5760Time, min from Epoch
Diff
eren
ce (m
)
NRLMSIS-00
J70
J71
J60
MSIS86 MSIS90
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Simulated Sensitivity Analysis• JERS sample orbit
– Different treatment of the data
– Baseline• Numerical
Propagation• Jacchia-Roberts• 3-hourly
geomagnetic
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
0 1440 2880 4320 5760Time, min from Epoch
Diff
eren
ce (m
)
Obs3HrSpl
Adj3HrObsConAllAvg
ObsConAllObsC81Dly
Obs3HrInt
ObsDly1700
ObsDly
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NRLMSISE-00 Results – Short Term
1.0
10.0
100.0
1000.0
1320 1560 1800 2040 2280 2520 2760 3000 3240 3480
Time, min from Feb 20, 2008 00:00:00.000 UTC
Diff
eren
ce (m
)
ObsConAllAvg
L81ObsConAll
LOS20CAD20
LOD20
LAI20
LAD20
COI20
COD20
CAT17CAT20
CAI20
CAS20
LOT20
LOS20COS20
LOI20
COT17
LAS20
COT20
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NRLMSISE-00 Results – Long Term
1.0
10.0
100.0
1000.0
10000.0
100000.0
0 1440 2880 4320 5760 7200 8640 10080 11520Time, min from Feb 20, 2008 00:00:00.000 UTC
Diff
eren
ce (m
)
ObsConAllAvgObsConAll
LOS20LAI20
CAT20CAT17
COS20
LAD20
COD20
CAD20
LAS20 CAI20
LAS20
LOT17LOT20
COT17COT20
CAS20
LOD20
• Observations:– Model specifies
observed• Adjusted
performed well– Centered 81-day
best– 20:00 UTC best– Spline
interpolation very good
• No single best answer
www.centerforspace.comPg 76 of 120
Jacchia-Roberts Results – Short Term
1.0
10.0
100.0
1000.0
1320 1560 1800 2040 2280 2520 2760 3000 3240 3480
Time, min from Feb 20, 2008 00:00:00.000 UTC
Diff
eren
ce (m
)
LOI20
ObsConAllAvg
L81ObsConAll
LOS20
CAI20 CAD20
LOD20LAI20
LAT17LAS20LAT20 LAD20
COI20
COD20
COT17COS20COT20
CAT20
CAS20
LOT17 LOT20
CAT17
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Jacchia-Roberts Results – Long Term
1.0
10.0
100.0
1000.0
10000.0
100000.0
0 1440 2880 4320 5760 7200 8640 10080 11520Time, min from Feb 20, 2008 00:00:00.000 UTC
Diff
eren
ce (m
)
LAI20
ObsConAllAvg
ObsConAll
LOS20
CAI20 CAD20
COD20COT20
• Observations:– Adjusted
performed well in all cases
– Centered 81-day best
– 20:00 UTC best– Daily
geomagnetic very good, but all were close
• No single best answer
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Atmospheric Drag Effects
• Atmospheric Drag– Large variations
• Changing the atmospheric model• Changing how the input data is interpreted
– F10.7 at 2000 UTC– Last 81-day average F10.7 vs. the central 81-day average– Using step functions for the atmospheric parameters vs
interpolation– Many others (see AIAA and UC paper)
– Point to take away: • 1-1000 km differences are possible• Unable to determine if from data interpretation or model
differences
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Third Body Forces
• Can affect GEO satellites strongly• Conservative force (like gravity)
)()(
33
33
3
3333
⊕
⊕
⊕
⊕⊕− −+
+−=
rr
rr
Gmr
rmmGa
sat
sat
sat
satsatbody
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Solar Radiation Pressure• Large effect for high altitude satellites (GPS, GEO, etc)
– Non conservative force• Shadowing by the Earth becomes very important
– All satellite altitudes• Solar Irradiance (psr) is difficult to measure accurately
Sunsat
SunsatSunRSRsrp r
rmAc
pa−
−−=
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Solar Irradiance (W/m2)
1360
1362
1364
1366
1368
1370
1372
1374
1376
1/1/1978 1/1/1980 1/1/1982 1/1/1984 1/1/1986 1/1/1988 1/1/1990 1/1/1992 1/1/1994 1/1/1996 1/1/1998 1/1/2000 1/1/2002
SMMNOAA-9NOAA-10NimbusCompositeURS2
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Earth Shadow Geometry
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Earth Shadow Geometry
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Solar Radiation Pressure Sensitivity Results
• Solar Radiation Pressure– Several variations shown
• 26690 (GPS)– Notice max is ~100m– Definitions
• Cylindrical– Defines shadow type
• App to true– Acct for light travel from
Sun to CB• True
– Inst light from Sun• No Boundary
– Change step size at penumbra/umbra
– Point to take away• Relatively small effect• Some variations
0.001
0.010
0.100
1.000
10.000
100.000
0 1440 2880 4320 5760
Time, min from Epoch
Diff
eren
ce (m
)
cylindrical
none
80.000
app to true
true
noboundary
www.centerforspace.comPg 85 of 120
Special Perturbations
• Numerically integrate the equations of motion– Time consuming, but accurate
othertidessrpbodydragsphericalnon aaaaaarr
ra ++++++= −− 33μ
www.centerforspace.comPg 86 of 120
General Perturbations
• Truncate analytical expansions and solve directly– Large time steps
• Each approach is mathematically different– SGP4– J2 only– Other
www.centerforspace.comPg 87 of 120
Semianalytical
• Blend numerical and analytical– Analytically solve secular and long period
components– Numerically integrate the small short period
variations• Fast and accurate
www.centerforspace.comPg 88 of 120
Force Model Sensitivity Results
• Force model contributions– Determine which forces contribute the largest effects
• 12x12 gravity field is the baseline
– Note• Gravity and Drag are largest contributors for LEO• 3rd body ~km effect for higher altitudes
– Point to take away:• Trying to get the last cm from solid earth tides no good unless all
other forces are at least that precise
www.centerforspace.comPg 89 of 120
Force Model Contributions
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
1000000.0
0 1440 2880 4320 5760
Time, min from Epoch
Diff
eren
ce (m
)
vs Two-Body
vs EGM-96 70x70
vs DragMSIS 00
vs DragJrob
vs ThirdBody
vs SRP
vs SolidTides
vsOceanTides
25544
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
1000000.0
0 1440 2880 4320 5760Time, min from Epoch
Diff
eren
ce (m
)
vs Two-Body
vs EGM-96 70x70
vs DragMSIS 00
vs DragJrob
vs ThirdBody
vs SRP
vs SolidTides
vsOceanTides
21867
• Low Earth OrbitISS JERS
www.centerforspace.comPg 90 of 120
Force Model Contributions
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
1000000.0
0 1440 2880 4320 5760Time, min from Epoch
Diff
eren
ce (m
)
vs Two-Body
vs EGM-96 70x70
vs DragMSIS 00
vs DragJrob
vs ThirdBody
vs SRP
vs SolidTides
vsOceanTides
7646
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
1000000.0
0 1440 2880 4320 5760
Time, min from Epoch
Diff
eren
ce (m
)
vs Two-Body
vs EGM-96 70x70
vs DragMSIS 00
vs DragJrob
vs ThirdBody
vs SRP
vs SolidTides
vsOceanTides
11
• Low Earth OrbitStarlette Vanguard II
www.centerforspace.comPg 91 of 120
Force Model Contributions
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
1000000.0
0 1440 2880 4320 5760Time, min from Epoch
Diff
eren
ce (m
)
vs Two-Body
vs EGM-96 70x70
vs DragMSIS 00
vs DragJrob
vs ThirdBody
vs SRP
vs SolidTides
vsOceanTides
22076
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
1000000.0
0 1440 2880 4320 5760
Time, min from Epoch
Diff
eren
ce (m
)
vs Two-Body
vs EGM-96 70x70
vs DragMSIS 00
vs DragJrob
vs ThirdBody
vs SRP
vs SolidTides
vsOceanTides
26690
• Low to Mid Earth OrbitTopex GPS
www.centerforspace.comPg 92 of 120
Force Model Contributions
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
1000000.0
0 1440 2880 4320 5760
Time, min from Epoch
Diff
eren
ce (m
)
vs Two-Body
vs EGM-96 70x70
vs DragMSIS 00
vs DragJrob
vs ThirdBody
vs SRP
vs SolidTides
vsOceanTides
25054
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
1000000.0
0 1440 2880 4320 5760Time, min from Epoch
Diff
eren
ce (m
)
vs Two-Body
vs EGM-96 70x70
vs DragMSIS 00
vs DragJrob
vs ThirdBody
vs SRP
vs SolidTides
vsOceanTides
20052
• Mid Earth Orbit, eccentricSL 12 RB Molnyia
www.centerforspace.comPg 93 of 120
Force Model Contributions
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
1000000.0
0 1440 2880 4320 5760
Time, min from Epoch
Diff
eren
ce (m
)
vs Two-Body
vs EGM-96 70x70
vs DragMSIS 00
vs DragJrob
vs ThirdBody
vs SRP
vs SolidTides
vsOceanTides
26038
0.1
1.0
10.0
100.0
1000.0
10000.0
100000.0
1000000.0
0 1440 2880 4320 5760
Time, min from Epoch
Diff
eren
ce (m
)
vs Two-Body
vs EGM-96 70x70
vs DragMSIS 00
vs DragJrob
vs ThirdBody
vs SRP
vs SolidTides
vsOceanTides
25544
• Low Earth and Geosynchronous OrbitISS (for comparison) Galaxy 11
www.centerforspace.comPg 94 of 120
• Fundamental Concepts• Newton• Kepler• Perturbations• Orbit Determination• Applications
Chapter 10
www.centerforspace.comPg 95 of 120
Terms
• Orbit Determination– Process of determining an orbit from observations– Also called Estimation
• Filtering– Determining the current state after each observation
• Smoothing– Improve previous state solutions using future data– Runs backwards
www.centerforspace.comPg 96 of 120
Terms
• Deterministic– Dynamics are known and can be calculated– Propagation
• Assuming a specific set of force models
• Stochastic– Uses observations to correct for unknown or
mis-modeled dynamics
www.centerforspace.comPg 97 of 120
Terms
• Least Squares– Minimizes the sum-square of the residuals– Depends on a fit span
• Length of time to process a batch of observations– Often called Batch Least Squares (BLS)
www.centerforspace.comPg 98 of 120
Linear Least Squares Example
• Assume a mathematical model of motiony = α + βx
• Residuals defined asri = yoi
– yci= yoi
– (α + β xoi)
• Cost function (Jacobian)
• Minimization of residuals
2 2
1 1
( , ) ( ( ))i i
N N
i o oi i
J r f y xα β α β= =
= = = − +∑ ∑
www.centerforspace.comPg 99 of 120
Linear Least Squares Example
• Matrix development
• Normal Equation– X = (ATA)-1ATb
11
21
1 2 1 2
1
11 1 ... 1 1 1 ... 11
... ...... ...1
N N
N
oo
oo
o o o o o o
oo
yxyx
x x x x x xyx
αβ
⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
AT ATA X b
www.centerforspace.comPg 100 of 120
Bias
Mean
Noise
True Position
.. . . .
... .
.
.. . . .
... .
.. . . .
... .
.. . . .
... .
.. . . .
... .
.. . . .
... .
.. . . .
... .
.. . . .
... .
.. . . .
... ... . . .
... .
.. . . .
... .
.. . . .
... ..
. . . .
... ..
. . . .
... ... . . .
... ..
. . . .
... .
.. . . .
... .
.. . . .
... .
.. . . .
... ..
. . . .
... .
. ..
.... . .
....
.. . . .
.. .
..
..
...
.. .
. ..
.... . .
....
.. . . .
.. .
.
.
..
...
.. .. .
...
.. . .
....
.. . . .
.. .
..
..
...
.. . . ..
.... . .
....
.. . . .
.. .
..
..
...
.. ...
...
.. . .
....
.. . . .
.. .
..
..
...
.. .
. ..
.... . .
....
.. . . .
.. .
..
..
...
.. .. .
...
.. . .
....
.. . . .
.. .
..
..
...
.. .. .
...
.. . .
....
.. . . .
.. .
..
..
...
.. .
..
...
.
..
.
...
.. . . .
... .
.. . . .
.. ..
....
..
.. ... ..... .
......... . .
..
....
.
..
.... . . .
....
.. . . .....
..
. .. ..
..
.. ..... .
......... .. .
..
....
... .. ... .. ...
Drift Noise
Drift ½ life
www.centerforspace.comPg 101 of 120
Statistical Concepts
• Dimensions and probability
2
1 exp( )2z−
−
( )2zerf
22( ) exp( )22
z zerf zπ
−−99.8997.0773.8519.873
99.9698.8986.4739.352
99.9999.7395.4568.271
4j3j2jz = 1jDimension
www.centerforspace.comPg 102 of 120
Covariance Matrix
• Measure of uncertainty• Grows with the satellite state propagation
P = (ATWA)-1
– W is weighting or sensor accuracies– A is partial derivative matrix
• Correlation Coefficients– Off diagonal terms
• Eigenvalues– Indicates each axis of the ellipsoid
www.centerforspace.comPg 103 of 120
LS Applied to Satellites: Overview
Orbit Determination
Propagate X to observation
timesloop through observations
Solve Jacobian
Form Residuals
Obtain Good Initial State Estimate, X
Initial Orbit Determination
How good?Radius of Curvature
What state representation?Equinoctial, Keplerian, other
How to solve for Jacobian?Analytical, finite differencing
Least Squares Solution method?Classical, Single Value Decomposition
Solve Least Squares
Converged?
www.centerforspace.comPg 104 of 120
LS Algorithm: Matrix Inverse Approach
• FOR i = 1 to the number of observations (N)– Propagate (SGP4, HPOP) nominal state to time of observation (TEME, ICRF)– Find the slant range vector, sensor to the propagated state in the topocentric (SEZ)
coordinate system – Determine nominal observations from the SEZ vector– Find the b matrix as observed – nominal observations– Form the A matrix
• Finite (or central) differences • Analytical partials
– H, Partials depending on observation type– Ф, Partials for state transition matrix.
– Accumulate ATWA and ATWb• END FOR• Find P = (ATWA)-1 using Gauss-Jordan elimination (LU decomposition and
back-substitution)• Solve δx = P ATWb• Check RMS for convergence• Update state X = X + δx• Repeat if not converged using updated state
www.centerforspace.comPg 105 of 120
Sequential Batch Least Squares
• Process additional observations– Use previous results
• Bayes Theorem
• Normal Equation– This is for “k” previously determined obs– “k + n” new obs
1 1
1 1
ˆ(0 | ) ( ) ( )ˆ ˆ ˆ(0 | ) ( )
T T Tnew new new k new new new k k k
Tk n new new new k
x k n A W A P A W b A W b
P P k n A W A P
δ − −
− −+
+ = + +
= + = +
www.centerforspace.comPg 106 of 120
Extended Kalman Filter
Actual Orbit
dx1
dx2 dx3dx4 dx5^
^ ^ ^ ^
X1_
^ P0
X0 ^
X1 ^
X2 ^
X3 ^
X4 ^
X5 ^
^ P2
^ P3
^ P4
^ P5
X2_
_P2
X3_
X4_
X5_
^ P1
_P3
_P4
_P5_
P1
www.centerforspace.comPg 107 of 120
Extended Kalman Filter
Predicted StatePredicted State ErrorPredicted Error Covariance
Kalman GainState Error EstimateError Covariance EstimateState Estimate
1
1
1
11
1
1 1
1
1
1 1 1
11 1 1 1 1 1
1 1
at each obs time ˆ
( , )
ˆ
ˆ
( , ) ( )
Prediction
U
( , )0
[ ]
pdate
ˆ
k
k kk
k
k
kk
t
k k t tt
t
t
k k k k
k
Tk k
k k k
T Tk k k k k k
k k k
zHX
X t t X dt X
XF
X
t t F t t tx
P P Q
b z H X
K P H H P H R
x x K
δ
δ δ
+
+
+
++
+
+ +
+
+
+ + +
−+ + + + + +
+ +
∂=
∂
= +
∂=
∂
Φ = Φ
=
=Φ Φ +
= −
= +
= +
∫
1 1
1 1 1 1 1
1 1 1
ˆ
ˆ ˆ
k
k k k k k
k k k
b
P P K H P
X X xδ
+ +
+ + + + +
+ + +
= −
= +
www.centerforspace.comPg 108 of 120
Averaging and Fit Spans• Obs are taken periodically• Updates often occur at regular intervals• Least Squares approaches “average” data collected for
a “batch” of time – the Fit Span
Time
Mean Change
Mean Change
Short-periodic plus long-periodic, and secular
Secular
Long-periodic andsecular
t1 t2 t3 t4
c
Time
Fit span, 3 days
Epoch 3Epoch 2Epoch 1
Obs
Daily Updates 1 2 3
www.centerforspace.comPg 109 of 120
• Fundamental Concepts• Newton• Kepler• Perturbations• Orbit Determination• Applications
Chapter 11
www.centerforspace.comPg 110 of 120
Applications• How do we put all this together and accomplish our
original goal?– Many analyses possible
• Prediction– Satellite look angles (Our original question)
• Behind the scenes– Time of observations– Coordinate systems throughout– Orbit determination of observations to obtain a state vector– Propagation to form an ephemeris– Calculations for Sun and Satellite to determine visibility– ...– And several other smaller details!
www.centerforspace.comPg 111 of 120
Satellite Orbital Characteristics
www.centerforspace.comPg 112 of 120
Predicting Satellite Look Angles
www.centerforspace.comPg 113 of 120
Rise Set Characteristics
www.centerforspace.comPg 114 of 120
Finding the Site Information (1)
• Approximate formulation– Non-rigorous ECEF– Don’t account for sidereal/solar time differences
ECIECIECI
SiteECIECIECI
SEZgdLSTECI
SEZgdLSTECI
rv
rr
ROTROT
ROTROT
~~~
~~
~
~
)]90(2)][(3[
)]90(2)][(3[
×+=
+=
−°−−=
−°−−=
⊕ωρρ
ρφθρ
ρφθρ
www.centerforspace.comPg 115 of 120
Finding the Site Information (2)
• Rigorous formulation (STK approach)– Precise ECEF– Account for sidereal/solar time differences
{ }PEFECEFTTTT
ECI
ECEFTTTT
ECI
pp
AST
TTUT
ECEFECEF
SiteECEFECEFECEF
SEZgdECEF
SEZgdECEF
rvPMSTNUTPRECv
rPMSTNUTPRECr
yROTxROTPMROTST
ROTROTROTNUTROTROTzROTPREC
TTTTTAIUTATUTUTCdaymonyrv
rr
ROTROT
ROTROT
×+=
=
−−==
ΔΨ−−=−Θ−=
⇒ΔΔ=
+=
−°−−=
−°−−=
⊕ω
θεεζ
ρρ
ρφλρ
ρφλρ
][][][][
][][][][
)(1)(2][)(3][
)(1)(3)(1][)(3)(2)(3][
),,,,1(),1,,,,(
)]90(2)][(3[
)]90(2)][(3[
1
Differences from Approximate
www.centerforspace.comPg 116 of 120
Results
• Rigorous approach– Position (ECI)
• -5505.504883 km• 56.449170• 3821.871726
• Simplified approach– Position (~ECI)
• -5503.79562 km• 62.28191• 3824.24480
• Difference– 6.52 km
• Perhaps this is acceptable?
www.centerforspace.comPg 117 of 120
Impact
• Applying textbook solutions to real-world problems will give the wrong answers– Assumptions add up– Examples:
• Communicating with a satellite using Laser comm– At orbital velocity, 2 sec is nearly 14 km
» Will your signal be able to locate and receive?• Will you pass System Acceptance Testing?
www.centerforspace.comPg 118 of 120
A word of Caution …
• Fundamentals vs Applications– Undergraduate vs Graduate– Classroom vs Operational– Attention to detail important– Nomenclature is important
www.centerforspace.comPg 119 of 120
Resources• Book
– Microcosm• Pam is here!
• http://www.celestrak.com/software/vallado-sw.asp– TLE data– EOP and Space Weather Data– Code
• SGP4• Other
– Errata• Not all updated but most are
– Solutions• Not complete – ask ☺
www.centerforspace.comPg 120 of 120
Questions??