C.K. Shum, Orbit Determination Notes 1 October 1997
NOTES ON ORBIT DETERMINATION MODELS
C.K. Shum, Ohio State University
October 1997
SPATIAL REPRESENTATION OF ALTIMETER CROSSOVERERROR DUE TO GEOPOTENTIAL PERTURBATION
Single satellite crossovers:
€
Δx = 2Δν
€
=2l =1
∞
∑m = 0
l
∑p=0
l
∑ lmpD lmps
Φ (C lm sinmλ−S lm cosmλ)
• Zonals unobservable (to this level of approximation)
Dual satellite crossovers:
€
Δx=Δyi −Δy j +Δvi −Δv j
€
=l=1
∞
∑m= 0
l
∑p= 0
l
∑ Dlmp˜ Φ lmp
c( ) i C lm cosmλ+ S lmsinmλ( )
€
−l=1
∞
∑m= 0
l
∑p= 0
l
∑ Dlmp˜ Φ lmp
c( ) i C lm cosmλ+ S lmsinmλ( )
€
ml=1
∞
∑m= 0
l
∑p= 0
l
∑ Dlmp˜ Φ lmp
s( ) i C lm sinmλ−S lmcosmλ( )
€
±l=1
∞
∑m= 0
l
∑p= 0
l
∑ Dlmp˜ Φ lmp
s( ) i C lm sinmλ−S lmcosmλ( )
for satellites i and j
C.K. Shum, Orbit Determination Notes 2 October 1997
SPATIAL REPRESENTATION OF THE RADIAL ORBIT ERRORDUE TO GEOPOTENTIAL PERTURBATION
For 0=q , radial orbit error [Tapley and Rosborough, 1985]
€
Δr(0) =l=1
∞
∑m= 0
l
∑p= 0
l
∑ lmpD lmp
c˙ Φ lmC cosmλ−lmS sinmλ( )
€
±l=1
∞
∑m= 0
l
∑p= 0
l
∑ lmpD lmp
s˙ Φ lmC sinmλ−lmS cosmλ( )
where
Dlmp - function of satellite altitude and inclination
Φ&c
lmp and Φ&
s
lmp - latitude functions
+ sign denotes satellite is on ascending pass
- sign denotes satellite is on descending pass
Geographical mean radial orbit error:
€
Δγ =l=1
∞
∑m= 0
l
∑p= 0
l
∑ lmpD lmp
c˙ Φ lmC cosmλ+lmS sinmλ( )
Geographical variability error about the mean:
€
Δv =±l=1
∞
∑m= 0
l
∑p= 0
l
∑ lmpD lmp
s˙ Φ lmC sinmλ−lmS cosmλ( )
C.K. Shum, Orbit Determination Notes 3 October 1997
ATMOSPHERIC DENSITY CORRECTION
€
ρ=ρm Δρ
where mρ - normal density model ρΔ - a correction factor
€
Δρ= oC 1+ 1C s− s s
solar flux effect
€
o 1+ 2C pK[ ] geomagnetic effect
€
o 1+ 3C cosφ cos3 Dlmp˜ Φ lmps( )−1 2( )[ ] diurnal bulge heating effect
€
o 1+ 4C oδεsinφ
seasonal-latitudinal effect
€
o 1+ 5C rro−1
altitude dependent
€
o 1+C 6sin ω + M( )+C 7cos ω + M( )[ ] once per revolution
€
o 1+∑ iA cos2πf it+ iB sin2πf it( )[ ] long period
SIMPLIFIED DENSITY CORRECTION MODEL
€
6C = 6C ⋅ DC
€
7C = 7C ⋅ DC
€
Δρ= DC +C 6sin M +ω( )+C 7cos M+ω( )
C.K. Shum, Orbit Determination Notes 4 October 1997
PRECISION ORBIT DETERMINATION METHODS
Dynamical Equations of Motion:
€
˙ ̇ r =− GMr r3 +∑ f r ,v ,c ,t( )
vr , - Position and Velocity Vectors
( )tcvrf ,,,∑ - Perturbation Forces
Gravitational:
• Non-spherical Earth• Luni-solar and planetary• Solid Earth tides• Ocean tides• General relativity
Nongravitational:
• Atmospheric drag• Direct solar radiation pressure• Earth albedo radiation pressure• Empirical forces
c - Constant Parameters• Dynamical• Kinematical
C.K. Shum, Orbit Determination Notes 5 October 1997
LEAST SQUARES SYSTEM SOLUTION TECHNIQUES
( ) yWHPHWHx TT 110
−−+=)
Conventional Approach:
• Gauss and Gauss-Jordan Elimination
• Gauss-Siedel Iteration
• Choiesky Decomposition
Orthogonal Matrix Approach:
• Householder Accumulation
• Gram-Schmidt Orthogonalization
• Givens-Gentleman Square-Root-Free Accumulation
C.K. Shum, Orbit Determination Notes 6 October 1997
GIVENS-GENTLEMAN ACCUMULATION TECHNIQUES
Find mmQ ×−− orthogonal matrix such that,
[ ]
=
ebR
yHWQ0
21
M
where
nnR ×− upper triangular matrix
1×−nb column vector
( ) 1×−− nme column vector
The resulting equation
bxR =)
can be solved by back-substitutions
C.K. Shum, Orbit Determination Notes 7 October 1997
ORDINARY DIFFERENTIAL EQUATION NUMERICALINTEGRATION TECHNIQUES
Single-Step Integrators
• Runge-Kutta 2(4), and 7(8) IntegratorFixed or variable step,For first-order equations
Multi-Step Integrators
• Sampine-Gordon IntegratorVariable order or variable mesh,For first-order equations
• Adams-Bashforth IntegratorFixed mesh and order,For first-order equations
• Krogh-Sampine-Gordon IntegratorFixed mesh and order,For second-order equations
C.K. Shum, Orbit Determination Notes 8 October 1997
N-BODY PERTURBATION
Expressing in a geocentric coordinate system, the central bodyand the N-body forces can be expressed as
Δ
Δ−
∇∑+∇
+= 31i
ii
i GMUGMU
MmF
where F is the force acting on the satellite due to N-bodyattraction; m and M are masses of the satellite and the Earth,respectively; iΔ is the position vector between the satellite andthe perturbing mass, iM ; and U∇ refers to the gradient of thegeopotential.
C.K. Shum, Orbit Determination Notes 9 October 1997
EARTH’S GRAVITY FIELD
02 =∇ U
( )[ ]λλ mSmCPrR
rGMU nmnmnm
ne
mnsincossin
00+Φ
= ∑∑
=
∞
=
where ( )ΦsinnmP are associated Legendre function of Φsin of degreen and order m ; nmC and nmS are normalized spherical harmoniccoefficients whose values are functions of Earth’s massdistribution; and eR is Earth’s mean equatorial radius. GM , nmC
and nmS can be parameters within the estimated state vector inthe orbit determination process.
C.K. Shum, Orbit Determination Notes 10 October 1997
SOLID EARTH TIDES
)()(12
2rV
rRkrU n
ne
nn
+∞
=
=Δ ∑
where )(rUΔ is the change in potential at position r , nk are Lovenumbers of degree n , and nV is the disturbing tidal potential ofdegree n .
The changes in geopotential caused by the luni-solar tides can beexpressed in terms of time-dependent geopotential coefficients;that is nmC and nmS which can be expressed as follows
211
)!()!)(2(4
+
−+=Δ
+
mnmnnq
GMRkC nm
nen
nm
211
)!()!)(2(4
+
−+=
+
mnmnnu
GMRkS nm
nen
nm
and
( ) jmjPmnmn
rGM
q nmnj
j
jnm Ψ
+
−=
+=∑ coscos
)!()!(2
11
3
2θ
( ) jmjPmnmn
rGM
u nmnj
j
jnm Ψ
+
−=
+=∑ sincos
)!()!(2
11
3
2θ
where the index 3,2,1=j denotes Earth, moon and sun.
C.K. Shum, Orbit Determination Notes 11 October 1997
OCEAN TIDES
The disturbing ocean tide potential, UΔ , can be expressed asfollows
( )⋅Φ
+
+∑∑∑∑=Δ
+−
+=
∞
=sin
1214
1'
00nm
nen
n
mnew P
rR
nkRGU
µρπ
( )±± ±±⋅ nmnm mtC µµµ ελβη ~)(sin~
where G is the gravitational constant; wρ is the mean density ofthe sea water; eR is the mean equatorial radius; '
nk is the loaddeformation coefficient for degree n ; m is the order of thecoefficient; µ is the ocean tide constituent index; ]'[)( 1pNphst τβ =
are the Doodson arguments which define lunar and solarephemeris and t is the time; ][ 621 nnnn ⋅⋅⋅= are integer multipliersof Doodson arguments; ±
nmCµ
~ are the amplitudes of ocean tideconstituents; and ±
nmµε~ are the phase angles.
Let
( )λβηµµ
µ
mtCSC
nm
nm
±⋅
=
±
±
±
)(cossin~
( )βηµ
µµ
⋅
−
+=
−+
−+
cosnmnm SS
CCBA
( )βηµ
µ
⋅
−
++
−+
−+
sinnmCC
SS
C.K. Shum, Orbit Determination Notes 12 October 1997
21
)2)(12()!()!(4
0
2
−+−
+=
m
enm nmn
mnM
wRFδ
ρπ
+
+
12'1
nk n
where M is the mass of the Earth; ±C , ±S are ocean tidecoefficients; and m0δ is the Kronecker delta function, 1=δ for 0=m ;
0=δ , otherwise.
The total potential, U that includes gravitational and tidalpotential and thus be expressed
( )
++
+
= ∑∑∑∑
=
∞
=
λµλµφµµ
mnmBFSmnmAFCPrR
rGMU nmnmnmnmnm
ne
n
mnsincossin
00
C.K. Shum, Orbit Determination Notes 13 October 1997
ATMOSPHERIC DRAG MODEL
Drag Acceleration of a Composite Satellite:
++−= nCvvCv
mA
vvmACa LD
pD 'cos 221 γρ
where
ρ - Atmospheric density
DC - Drag coefficient for the satellite main body
'DC and LC = Drag and lift coefficients for the satellite solar panels
A and pA - Cross-sectional areas of main body and solarpanels perpendicular to v
v - Velocity of satellite relative to the atmosphere
γcos = nvv⋅
n - Unit vector normal to the solar panels
m - Satellite mass
Semi-Empirical Thermospheric Density Models
• Jacchia Models (1965, 1971, 1977)• Mass spectrometer-Incoherent Scatter Models (MSIS)• Drag Temperature Model (DTM)• ESRO-4 Models
C.K. Shum, Orbit Determination Notes 14 October 1997
CHARACTERISTICS OF UPPER ATMOSPHERE
Diurnal Variation
• Solar heating, atmospheric bulge
Solar Activities
• 27-day variation: solar rotation
• 11 year: solar cycle
• Solar decimeter flux (10.7 cm wavelength)
Geomagnetic Activities
• Geomagnetic index ),( pp AK
Seasonal-Latitudinal
• Seasonal migration of Helium
Semi-Annual Variation
• Year to year, vary in amplitudes
• Probably not due to solar activities
Others
C.K. Shum, Orbit Determination Notes 15 October 1997
ATMOSPHERIC DENSITY CORRECTION
ρρρ Δ= m
where mρ - normal density model ρΔ - a correction factor
−+=Δ
sssCCo 11ρ solar flux effect
[ ]KC p21+o geomagnetic effect
( )( )[ ]21~coscos1 33 −Φ+ s
lmplmpDC φo diurnal bulge heating effect
+ φ
εδ sin1 4
oCo seasonal-latitudinal effect
−+ 11 5
orrCo altitude dependent
( ) ( )[ ]MCMC ++++ ωω cossin1 76o once per revolution
( )[ ]tftf iiii BA ππ 2sin2cos1 +∑+o long period
SIMPLIFIED DENSITY CORRECTION MODEL
CCC D⋅= 66
CCC D⋅= 77
( ) ( )
++++=Δ ωωρ MM CCCD cossin 76
C.K. Shum, Orbit Determination Notes 16 October 1997
VALIDATION OF DENSITY CORRECTION MODEL
DENSITY CORRECTION MODEL STUDY
Simulation Summary
SEASAT 9-day orbit trajectory fit Jacchia 1971 drag model filledwith nominal exponential model estimating density correction
parameters.
Estimated Parameters Along-Track RMS (m) Radial RMS(m)
Global DC 9.18 0.67Daily DC 1.37 0.66Global 321 ,, CCC and 4C 6.04 0.443-Day 321 ,, CCC and 4C 0.48 0.02Daily DC , 3-Day 6C and
7C0.36 0.02
C.K. Shum, Orbit Determination Notes 17 October 1997
DENSITY CORRECTION MODEL ON STARLETTE ORBIT
Arc Length: March 18 – April 17, 1983
30-day arc
Date Type: Laser Range
Density Model: DTM
Estimated Parameters: 6,,,, CCCvr RDrv and 7C
Case Density Model Correction Range RMS (m)
1 10-day DC 127.0
2 10-day DC , 30-day 6C and 7C 111.0
3 10-day DC , 15-day 6C and 7C 83.6
4 5-day DC , 15-day 6C and 7C 78.5
C.K. Shum, Orbit Determination Notes 18 October 1997
DENSITY CORRECTION MODEL ON SEASAT ORBITS
Arc Lengths: July 28 – August 14, 198518-day arc
September 15 – October 8, 198524-day arc
Data Weights: Laser range - 1 mUnified S-Band - 1 cm/secTranet doppler - 1 cm/sec
Estimated Parameters: 76 ,,,,, CCCCvr RDrv and resonant geopotentialcoefficients
RMS
Case Arc MeasurementsUsed
DensityModel
Laser(cm)
USB(cm/s)
Tranet(cm/s)
Crossover(cm)
1 18day
USB, laser DTM, dailyDC
75.3 1.41 -- 92.9
2 18day
USB, laser DTM, dailyDC , 6C and
7C
73.3 1.39 -- 89.8
3 18day
USB, laser Exponential,_-day DC ,
6C and 7C
67.2 1.39 -- 95.9
4 24day
USB, laserand Tranet
J71, _-dayDC
77.0 1.58 1.59 80.0
5 24day
USB, laserand Tranet
J71, _-dayDC , 6C and
7C
75.8 1.56 1.27 76.0
C.K. Shum, Orbit Determination Notes 19 October 1997
SATELLITE RADIATION PRESSURE
[ ]dsunIFs
θβγθ cos)1()f( −+∫∫−=
n = unit vector normal surface element ds
u = unit vector along satellite-sun direction
θ = angle between n and u
I = incident flux ~1350 watts/m2
[ ] θθγβγθβγθ cos)()1()1((f)cos2)f( 2 KB −+−+=
γ = total reflectivity
β = specular reflectivity
(f)B = diffuse reflection and reradiation function
= 2/3 for Lambert’s cosine law
TTTT
bbff
bbffK 44
44
)(εε
εεθ
+
−=
bf εε , = front and back emissively
bf TT , = front and back temperature
from Georgivic, 1976.
C.K. Shum, Orbit Determination Notes 20 October 1997
CONVENTIONAL RADIATION PRESSURE MODELS
uAIF )1( η+−=
I = incident flux ~ 4.5 x 10-6 N/m2
A = cross-sectional area (may vary)
u = satellite-sun unit vector
η = “reflectivity” coefficient includes lumped effect ofspecular and diffuse reflection and unbalanced thermalradiation
All forces assumed to be along u ; off axis forces assumednegligible or having zero mean.
Solar panels can be modeled more exactly given knowledge oforientation, but approximations are still made about specular vs.diffuse reflection and about temperature imbalance.
For Topex:
A ~ 7 m2 (ignoring panels)
η ~ 0.2 – 0.5 (lumped reflectivity/emissivity)
Acceleration mFa = , mass = 2000 kg
a ~ 2-5 x 10-8 m/sec2
Therefore, 1% imbalance ≥ 2 x 10-10 m/sec2
C.K. Shum, Orbit Determination Notes 21 October 1997
Simplifying Hall and Fote [1981] and ignoring internallygenerated heat, surface temperature is determined by
4)()/()( TAtdtdTTCV σεφα −=
T = surface temperature
)(TCV = heat capacity
α = absorptivity
ε = hemispherical emittance
σ = Stefan-Boltzman constant
A = surface area
)(tφ = radiant power = θsinEA
E = solar irradiance (~1350 watts/m2)
θ = angle of irradiance
C.K. Shum, Orbit Determination Notes 22 October 1997
EARTH RADIATION PRESSURE MODEL
Earth Radiation Pressure is the radiation pressure exerted on asatellite due to:
• Sunlight reflected off the earth (shortwave)
• Heat emission of the earth (longwave
THINGS THAT “LOOK” LIKE EARTH RADIATION PRESSURE
• Solar Radiation Pressure Reflectivity
• Odd Zonal Geopotential Coefficients
• Some Tide Coefficients
• Drag Coefficients (under some circumstances)
C.K. Shum, Orbit Determination Notes 23 October 1997
MODELING THE TERRESTRIAL SPHERE
I. Nominal Diffuse Model
• Earth-atmosphere system assumed to be purely diffusereflector and emitter
• Albedo and longwave emissivity vary with:
o Latitude )(sin)(sin)( 2210 1 φφ PaPtaaa ++=
o Season )sin()cos()( 2101 tctccta ωω ++=
II. Enhanced Diffuse Model:
• Longitudinal albedo variations included. (4 x 4 sphericalharmonic expression)
III. Advanced Earth Radiation Pressure model:
• Anisotropy in reflected component included.
• Diurnal variations in albedo included.
C.K. Shum, Orbit Determination Notes 24 October 1997
EARTH ORIENTATION AND TIME MODEL(For updated models see IERS Standards 2005)
Conventional Inertial System (CIS):
J2000
Conventional Terrestrial System (CTS):
Lageos-derived system (CSR 2000) [Richard Eanes, 2000]
Precession:
1976 IAU formula [Lieske, 1977]
Nutation:
1980 IAU nutation formula [Wahr, 1981b; Seidelman, 1982]
Polar Motion:
Smoothed values from Lageos earth rotation series (CSR 2000)
UT1-TAI
Smoothed values from Lageos plus short-period (<40 days) tidalvariations from Yoder et al. [1981]
C.K. Shum, Orbit Determination Notes 25 October 1997
FORCE MODEL SUMMARY: GRAVITATIONAL
Earth*:
EGM-96, TEG-4, GGSM’s, EIGENGM = 398600.4414 km3 s-2
ae = 6378136.50 m
Luni-Solar:
Point mass sun, moon and planets with DE-200 ephemeris[Standish, 1982]
Solid Tides:
Neutral 1066 A earth model [Gilbert and Dziewonski, 1975] usingWahr [1981a] formulation
Frequency independent k2 = 0.3 and δ = 0 used
Honkasalo correction to C20 applied, 820 10391.1 −×−=ΔC per unit k2
Ocean Tides:
Schwiderski [1980] based on MERIT Standards [Melbourne et al.,1983]
General Relativity:
One-body (earth), 1== βγ
* GSFC, Texas, GFZ Gravity models
C.K. Shum, Orbit Determination Notes 26 October 1997
FORCE MODEL SUMMARY: NONGRAVITATIONAL
Atmosphere:
Drag Temperature Model (DTM) density model [Barlier, 1977],DTM200, MSIS, etc. with daily solar flux and geomagnetic indices[Solar-Geophysical Data, Boulder, NOAA]
Cross-sectional area model: Constant or area variation, mass andarea are satellite dependent
Solar Radiation Pressure:
Conical shadow model of radius 6371 km
Constant cross-sectional area model
Earth Radiation Pressure:
Zonal and periodic terms for earth albedo and emissivity
C.K. Shum, Orbit Determination Notes 27 October 1997
LASER MEASUREMENT MODEL
Model:
Instantaneous range
Laser Station Coordinates:
Lageos-derived coordinates CSR 2002 [R. Eanes, 2000]
Earth Tides:
h2 = 0.609, l2 = 0.0852, δ = 0 [Farrell, 1972]
(no ocean loading)
Center of Mass:
Correction from reflector to the center of mass (satellitedependent)
C.K. Shum, Orbit Determination Notes 28 October 1997
ALTIMETER MEASUREMENT MODEL
Model:
Ocean surface model composed of gravitational and rotationalpotential, Schwiderski [1980] ocean tide model and Levitus[1982] sea surface topography model.
Gravitational Potential:
TEG-4 (200x200) [Tapley et al., 2002]
oW = 6263861 m2/sec2
GM = 3698600.4414 km3/sec2
ea = 6378137 m
Ocean Tide Model:
Dynamical contribution using MERIT ocean tide only [Melbourne,et al., 1983]. Geometric correction at the ocean surface used thefull model Schwiderski [1980].
Sea Surface Topography Model:
Spherical harmonic representation (10x10) computed by Engelis[1985] based on the model developed by Levitus [1982]
Data Smoothing*:
Raw altimeter data smoothed to represent spectrum consistent todegree and order 50* Not implemented for the current gravity field solution
C.K. Shum, Orbit Determination Notes 29 October 1997
ALTIMETER MEASUREMENT MODEL
Observed:
nbhhhhh SSTacs ++++++= ζ
- sh is the instantaneous distance between the satellite altimeterand the ocean surface.
- nbhh ac ,,, are corrections and random error.
- SSTh is the sea surface topography ( SSTh can be represented by atime series).
( ) CSC lmmmml
SST Pmmh '
00
''
01)(sinsincos ++= ∑∑
−=
φλλ
- ζ is the ocean tide height
)),(cos(),( λφδχλφζζ kkkkk
−+Θ=∑
[ ] )(sin)sin()cos(00
φλλ lmkklmkklm
l
mlkPmm SC ±Θ+±Θ=
±±−
+=
∞
=∑∑∑∑
- kΘ , kχ are the Doodson argument and phase correction forconstituent k
- ),( λφζ k is the amplitude of the tide height at geographic location),( λφ
- ),( λφδ k is the phase of the tide height at geographic location ),( λφ
- SCSC klmklmklmklm
−−++ ,,, are the unnormalized prograde and retrogradetide coefficients
C.K. Shum, Orbit Determination Notes 30 October 1997
ALTIMETER MEASUREMENT MODEL
(continued)
Computed:
ec rrh rrr−=
- err can be computed from:
€
w=wo ; ∇w∇w
=) u h
- w is the potential for the global ocean surface, ow is a constant
- w∇ is the gradient of the potential, w , and is normal to w .
- hu) represents the unit vector for the altimeter measurement.
Observation Equation:
chhyr
−=
Dynamically consistent model if w is used as the perturbingpotential affecting the satellite’s orbit.