Upload
others
View
16
Download
1
Embed Size (px)
Citation preview
Satellite Altimetry andSatellite Altimetry andGravimetryGravimetry: : Theory and ApplicationsTheory and Applications
C.K. ShumC.K. Shum1,21,2, Alexander Bruan, Alexander Bruan2,12,1
1,21,2Laboratory for Space Geodesy & Remote SensingLaboratory for Space Geodesy & Remote Sensing 2,12,1Byrd Polar Research CenterByrd Polar Research Center
The Ohio State UniversityThe Ohio State UniversityColumbus, Ohio, USAColumbus, Ohio, USA
[email protected]@osu.eduedu, , [email protected]@osuosu..edueduhttp://geodesy.eng.ohio-state.http://geodesy.eng.ohio-state.eduedu
Norwegian Univ. of Science and TechnologyTrondheimTrondheim, Norway, Norway
2121––25 June, 200425 June, 2004
Satellite Altimetry andSatellite Altimetry and Gravimetry Gravimetry::Theory and ApplicationsTheory and Applications
C.K. Shum & Alexander Braun,C.K. Shum & Alexander Braun, Ohio State UniversityOhio State UniversityContributors:Contributors:K. Cheng, Shin-K. Cheng, Shin-chan chan Han, Chung-yenHan, Chung-yen Kuo Kuo, , Yuchan Yuchan Yi, Ohio State Yi, Ohio State UnivUniv..Acknowledgements:Acknowledgements:Brian Beckley, NASA/GSFC Altimeter Pathfinder Data CenterBrian Beckley, NASA/GSFC Altimeter Pathfinder Data CenterJérôme Benveniste, Pierre Femenias, ESA/ESRINDudley Dudley CheltonChelton, Oregon State University, Oregon State UniversityDon Chambers, JohnDon Chambers, John Ries Ries, Bob , Bob SchutzSchutz, Byron , Byron TapleyTapley, , UnivUniv. of Texas. of TexasCheinway Cheinway Hwang, National Hwang, National Chiao Tung Chiao Tung University, TaiwanUniversity, TaiwanJohn John LillibridgeLillibridge, NOAA/Lab. for Satellite Altimetry, NOAA/Lab. for Satellite AltimetryLee Fu, Margaret Lee Fu, Margaret SrinivasanSrinivasan, Robert , Robert BenadaBenada, NASA/JPL, NASA/JPLPhilip Woodworth,Philip Woodworth, Proudman Proudman Oceanographic LaboratoryOceanographic Laboratory*Last Lectures:*Last Lectures: Wuhan Wuhan Altimetry Workshop, 2002; GLOSS Malaysia Workshop, 2004Altimetry Workshop, 2002; GLOSS Malaysia Workshop, 2004
Satellite Altimetry andSatellite Altimetry and Gravimetry Gravimetry::Theory and ApplicationsTheory and Applications
•• Orbital Dynamics and Orbit Determinations I Orbital Dynamics and Orbit Determinations I (AM) By C.K. Shum(AM) By C.K. Shum–– Keplerian Keplerian motion, general perturbation, motion, general perturbation, KaulaKaula’’s s formulationsformulations–– Periodic variations and resonances due to Periodic variations and resonances due to geopotentialgeopotential
•• Introduction to Satellite Altimetry I Introduction to Satellite Altimetry I (PM) By Alexander Braun(PM) By Alexander Braun–– What is altimetry?What is altimetry?–– Basic principles of satellite altimetry and its historyBasic principles of satellite altimetry and its history–– Interdisciplinary applications of altimetryInterdisciplinary applications of altimetry
•• Tutorial on Tutorial on iGMT iGMT –– a graphics tool a graphics tool (PM) By Alexander Braun(PM) By Alexander Braun
Monday, 21 June 2004Monday, 21 June 2004
Satellite Altimetry andSatellite Altimetry and Gravimetry Gravimetry::Theory and ApplicationsTheory and Applications
Tuesday, 22 June 2004Tuesday, 22 June 2004•• Orbital Dynamics & Orbit Determinations IIOrbital Dynamics & Orbit Determinations II (AM) By C.K. Shum(AM) By C.K. Shum
–– Nonlinear orbit determination & parameter recoveryNonlinear orbit determination & parameter recovery–– Force, measurement, and Earth orientation models for PODForce, measurement, and Earth orientation models for POD
•• Satellite Altimetry IISatellite Altimetry II (PM) By C.K. Shum(PM) By C.K. Shum–– Principles of satellite altimetry, mission design, waveformsPrinciples of satellite altimetry, mission design, waveforms–– Altimeter crossovers, geographically correlated orbit errorsAltimeter crossovers, geographically correlated orbit errors–– Precision orbit determination and accuracy evaluationsPrecision orbit determination and accuracy evaluations–– Instrument, media and geophysical correctionsInstrument, media and geophysical corrections
•• Tutorial onTutorial on iGMT iGMT (continued:) (continued:) (PM) By Alexander Braun(PM) By Alexander Braun
Satellite Altimetry andSatellite Altimetry and Gravimetry Gravimetry::Theory and ApplicationsTheory and Applications
Wednesday, 23 June 2004Wednesday, 23 June 2004•• Space Geodesy: An Interdisciplinary ScienceSpace Geodesy: An Interdisciplinary Science (AM) (AM) C.K. ShumC.K. Shum
–– Nonlinear orbit determination & parameter recoveryNonlinear orbit determination & parameter recovery–– Example interdisciplinary applications of satellite geodesyExample interdisciplinary applications of satellite geodesy
•• Altimeter Collinear Analysis Altimeter Collinear Analysis (PM) By Alexander Braun(PM) By Alexander Braun–– Stackfile Stackfile method for oceanography and marine geophysicsmethod for oceanography and marine geophysics–– Mean sea surface, marine gravity field determinationsMean sea surface, marine gravity field determinations–– Models and accuracy evaluations and limitationsModels and accuracy evaluations and limitations
•• Radar Altimeter Data Processing Radar Altimeter Data Processing (PM) By Alexander Braun(PM) By Alexander Braun–– Demonstration of GFZ ADS systemDemonstration of GFZ ADS system–– Other data centers (access, data ordering, data products)Other data centers (access, data ordering, data products)
Satellite Altimetry andSatellite Altimetry and Gravimetry Gravimetry::Theory and ApplicationsTheory and Applications
•• Themes:Themes:–– Orbital dynamics and orbit determinationOrbital dynamics and orbit determination–– Instrument error budget and analysisInstrument error budget and analysis–– Geophysical inverse programGeophysical inverse program–– Interdisciplinary applicationsInterdisciplinary applications
•• Basic knowledgeBasic knowledge–– Orbital mechanics, dynamics, physical and satellite geodesyOrbital mechanics, dynamics, physical and satellite geodesy–– Mathematical tools (linear algebra, statistics, numerical analysis,Mathematical tools (linear algebra, statistics, numerical analysis,
differential equations, approximate theory, adjustment)differential equations, approximate theory, adjustment)–– Physics, astronomy, engineeringPhysics, astronomy, engineering–– Instrument and their principles (radar, optical, Instrument and their principles (radar, optical, electromagneticselectromagnetics))–– Geophysics, oceanography, atmosphere, hydrology, glaciologyGeophysics, oceanography, atmosphere, hydrology, glaciology
Satellite Altimetry andSatellite Altimetry and Gravimetry Gravimetry::Theory and ApplicationsTheory and Applications
•• A good geodesist A good geodesist shouldshould::–– Know assumptions made in theory and data processingKnow assumptions made in theory and data processing–– Know your instrument well (Know your instrument well (precision and accuracyprecision and accuracy))–– Define your problem to solve first, then the correspondingDefine your problem to solve first, then the corresponding
observation requirementsobservation requirements–– Always provide Always provide real real uncertainty of your uncertainty of your solutions solutions or or productsproducts–– Know that the knowledge and instrument accuracy are Know that the knowledge and instrument accuracy are movingmoving
targets that changes with timetargets that changes with time–– Always question your instructorsAlways question your instructors’’ lectures: they could be wrong lectures: they could be wrong–– Look for new problems not necessarily in your field to be solvedLook for new problems not necessarily in your field to be solved
using your instrumentusing your instrument
Satellite altimetry measures with an accuracy of 1 part perSatellite altimetry measures with an accuracy of 1 part perbillion: few cm accuracy at ~1500 km altitude to center of Earthbillion: few cm accuracy at ~1500 km altitude to center of Earth
Orbital DynamicsOrbital DynamicsC.K. ShumC.K. Shum1,21,2, Shin-, Shin-chanchan Han Han11
1,21,2Laboratory for Space Geodesy & Remote SensingLaboratory for Space Geodesy & Remote Sensing 2,12,1Byrd Polar Research CenterByrd Polar Research Center
The Ohio State UniversityThe Ohio State UniversityColumbus, Ohio, USAColumbus, Ohio, USA
[email protected]@osu.eduedu, , [email protected]@osuosu..edueduhttp://geodesy.eng.ohio-state.http://geodesy.eng.ohio-state.eduedu
Norwegian Univ. of Science and TechnologyTrondheimTrondheim, Norway, Norway
2121––25 June, 200425 June, 2004
Original Figure from K. Lambeck
Useful quantities/constantsSome fundamentals: Timesystem, etc
Some Useful Geodetic Constants & AccuraciesEarth Rotation and Polar Motion Accuracy1960: x, y, 30 mas (1 ppb ~ 6 mm)2000: x, y, 0.2 mas; UT1: 20 µsec; Scale, 1+0.7 ppbGravity1 gal ~ 1 m/sec2; 1 µgal (change) ~ 5 mm (surface of the Earth)1 E = 1 mgal/10 km; 1 mE = 10-12 s-2
GOCE: ∇∇TU ~ 2nd order tensor; GRACE: δ∇U ~ δ potentialGOCE Gradiometer Accuracy: 2 mE/Hz1/2 (0.005 Hz to 0.1 Hz)GRACE Inter. ranging Acc.: 0.1–0.5 µm/s; 5 cm geoid (>250 km)1 µradian (µrad) = 1 mm/kmOtherRadar: Ku-band (13.4 GHz): 2.8 cm; C-band (5.6 GHz) 5.6 cm;Ionosphere: 1 TEC = 2.2 mm@Ku band; 16 cm@L-bandSea level: 1–2 mm/yr ~ 400 Gton/yr; 1 mm/yr = 0.2 µgal/yr
Determination of Time: BasicsTime System Conversions:
Concept_1 Sidereal day = o36024 =h
In 1995, 1 mean solar day day siderealmean 9351.00273790=
timesidereal of 55537.56324 smh ×= o36050027379093.1 ×=
0 12UT
equal to how many degree
o
L
3601.0027370.5
day sidereal1.0027370.5
daysolar 0.5UT12
××=
×=
=
Courtesy: C. Hwang
Universal Time
Mean sun: an imagined sun with a uniform motion on theplane of the equatorMean solar time: hour angle of the mean sunMean solar day: time interval between two successivetransits of the mean sunUniversal time (UT): Greenwich hour angle of the mean sun h12+
€
UT12h=12h + Greenwich hour angle of the mean sun
UT0 = UT as deduced directly from observationsUT1= UT0 + Λp(polar motion), represents the true angular rotation of the earth (i.e., includes tidal variations)
Conversion from UT1 to GAST_
€
GAST =1.00273790935UT1+ϑ 0 + Δψ cos∈ (1.37) ϑ 0 = 24110•
s54841+ 8640184•s812866T + 0•
s093104T 2 - 6•s2 ×10−6T 3 (1.38)
T = (tu − 2451545) /36525tu = Juliandays at 0h UT1
Determination of Time: Basics
Courtesy: C. Hwang
1.2.3 Dynamical Time
TDB: Barycentric Dynamic Time_Time derived from orbital motionsreferred to the barycenter center (center of mass) of the solarsystem. The coordinate time.
TDT: Terrestrial Dynamical time, referred to geocenter. The proper time
1.2.4 Atomic times
TAI: International atomic time, a weighted mean of individual clocksfrom around the word, in SI(International System of Units) second.
UTC: Coordinated Universal time, is adapted to UT1 by “leap seconds”
(1.40) 0.7sQUT1UT1-UTC
(1.39) (1s)n-TAIUTC
≤=
×=
UTC is the Broadcast Time in most countries (with leap second)
Determination of Time: Basics
Courtesy: C. Hwang
GPST: Known also as GPS Time, which is one kind of atomic time, but isused and adopted by the GPS control system for time transfer.
s
s
90
90
0
10232C Jan,1,1992
101376C Jan,1,1989
(1.41) BIPM from integer, n ; C-snUTC-GPST
−
−
×=
×−=
×=
M
Other relationships_
offsetconstant 18432-TDTTAI
offsetconstant 00019GPSTTAIs.
s.
=
+= (1.42)
Determination of Time: Basics
Courtesy: C. Hwang
1.2.5 Calendar
Julian date (JD): number of mean solar days elapsed since the epoch 4713 B.C.,January,1.5 Modified Julian date (MJD): JD–2,400,000.5 FK5 standard epoch: 2000 January 2,451,545JD ,51. =d
GPS standard epoch: 1980 January 52,444,244.JD,06. =d
MJD for 0.2000J : 51,544.5
Determination of Time: Basics
Courtesy: C. Hwang
Orbital Dynamics:Keplerian motion, Kepler’sequations
1. Dynamics of Satellite Orbits 1.1 Elliptic (Keplerian) Motion
Satellite's motion = Elliptic motion + Perturbed motion
(1) => and are co-linear. The double integration leads to six integration parameters spanning an orbital plane.
They are / or .
/ is called a state vector and is called the Keplerian orbital elements.
a – semi-major axis of orbital ellipse, e – eccentricity of orbital ellipse, i – inclination, ω – argument of perigee, Ω – right ascension of ascending node, ν – true anomaly.
GM , 3
=−= µµxx
r&&
x&& x
x&
x
),,,,,( νωΩiea
x&
x
),,,,,( νωΩiea
Notes from S. Han
Satellite Orbit Geometry and Orbit ElementsSatellite Orbit Geometry and Orbit ElementsX, Y, Z: non-rotating system (J2000) or theConventional Inertial System, or CIS,(An approximation to the inertial system)
a – semi-major axis of orbital ellipse e – eccentricity of orbital ellipse i – inclination ω – argument of perigee Ω – right ascension of ascending node ν – true anomaly
io
h is the angularmomentum vector
(a, e) – size and shape of the orbit(i, ω, Ω ) –orientation of orbit wrt CTS(ν ) – time dependent variable
1. Dynamics of Satellite Orbits 1.1 Elliptic (Keplerian) Motion
For the parameter indicating the orbiting particle's location on the orbital ellipse, we can use otherquantities such as eccentric anomaly, E, or mean anomaly, M. Let us introduce the orbital ellipseas follows:
Notes from S. Han
1. Dynamics of Satellite Orbits 1.1 Elliptic (Keplerian) Motion
From (6), we have the angular momentum conservation equation as follows:
(7) Combining this with (5), the equation of the ellipse can be derived as follows:
(8) From the above geometry, we realize the followings:
(9) (10)
Consequently, the distance can be written in terms of the eccentric anomaly as follows:
(11) From (7), (9), and (10) we have (12)
(13)
( ) .1 22 consteah =−== µνρ &
ννρ
cos1)1(
)(2
eea
+−
=
( ) Eea sin1sin 2−=νρ
aeEa −= coscosνρ
( )EeaE cos1)( −=ρ
( ) ννρ
ρ dea
ed sin
1 2
2
−=
EdEaed sin=ρNotes from S. Han
1. Dynamics of Satellite Orbits 1.1 Elliptic (Keplerian) Motion
(14) After some mathematical manipulation, we will have
(15) where The integration of (15) gives Kepler’s Equation:
(16) where , which is called mean anomaly. In summary, three anomalies (true, eccentric, mean), ν(t), E(t), M(t), of the Keplerian orbit andtheir relationships are given as follows:
(17)(18)
(19)
( )dtead 22 1−= µνρ
( ) ndtdEEe =− cos1
3an
µ=
MEeE =− sin
( )0ttnM −=
( )0)( ttntM −=)(sin)()( tEetMtE +=
−
+−=
Ee
Eet
cos1
cosarccos)(ν
Notes from S. Han
1. Dynamics of Satellite Orbits 1.1 Elliptic (Keplerian) Motion
In order to describe the motion of the satellite on the orbital ellipse in the inertial coordinatesystem, let us introduce the coordinate system attached to the orbital ellipse. The first coordinatepoints the perigee, the third coordinate is normal to the orbital plane, and the second coordinate isautomatically decided by the right handed rule of the coordinate system. In this coordinatesystem, we have:
(20)
Taking derivatives of each component, we have the velocity vector as follows:
(21)
−
−
=
=
0
sin1
cos
0
sin
cos2 Ee
eE
ar ν
ν
r
−
−
=
+
−
−=
0
cos1
sin
0
cos
sin
1
22
2Ee
E
r
nae
e
naν
ν
r&
Notes from S. Han
1. Dynamics of Satellite Orbits 1.1 Elliptic (Keplerian) Motion
Using both vector representations, we can derive the energy conservation law and angularmomentum conservation law as follows:
(22) (23)
where h is the constant angular momentum vector orthogonal to the orbital plane The first equation indicates Energy Conservation:
The second equation indicates Angular Momentum Conservation. In the Two-Body program, theorbital plane in invariant. This means that the inclination or the orientation of the orbital planedoes not change. In the real world, it is a good approximation.
€
12ddt
˙ r ⋅ ˙ r ( ) − µr
= const.
hrr =× &
€
v22−
µr
=µ2a
Notes from S. Han
1. Dynamics of Satellite Orbits 1.1 Elliptic (Keplerian) Motion
The state vector in the inertial coordinate frame can be obtained by applying appropriate rotationsto and . From the geometry shown in the previous figure, it can be done as follows:
(24) (25)
where For the conversion algorithm between Kelperian elements and a state vector, we can referMcCarthy et al. (1993) and Seeber (1993).
( ) ( ) ( ))or , ,(,,,,,,,,, νωω EMeaiMiea rRx ⋅Ω=Ω
( ) ( ) ( ))or , ,(,,,,,,,,, νωω EMeaiMiea rRx && ⋅Ω=Ω
( ) ( ) ( ) ( )ωω −−Ω−=Ω 313,, RRRR ii
Notes from S. Han
Example Problems 1.1 Elliptic (Keplerian) Motion
The maximum latitude of the groundtrack of a retrograde circular orbitis 710. The satellite crossed the equator heading north at longitude2200E on its first revolution and at longitude 1900E on the nextrevolution. What are the inclination and the altitude of this satelliteorbit? (Hint: assume Keplerian motion, use µ⊕=3.986x105 km3/s2,Re=6378 km, ω⊕=360.9856 0/day).
Solution
For retrograde orbit, i = 1800 – 710 = 1090
In one orbital period, the change in longitude is: Δλ = 1900 – 2200 = –300
The rate of rotation of the Earth, R, is: 3600/sid. Day x 366.2422/365.2422 sid.day/solar day = 360.98560/day
Now, Δλ = –RxTP
TP = Δλ/(–R) = 7180.34 s
TP = 2π a 3/2 / µ1/2
Therefore, a = 8044.3 km or Altitiude = a – 6378 km = 1666.3 km
Example Problems 1.1 Elliptic (Keplerian) Motion
Kepler’s Equation.
Show that the Earth has approximately 185 days in each year during which its distance isfarther than 1 Astronomical Unit (AU) from the Sun. (Note: 1 AU is the mean distancethe mean semi-major axis, a, of the Earth’s orbit around the Sun, e=0.00167).
Hint: Use the following equations:
M = E − esin E
Δt = (M2 − M1) /n
n = 3600 / 365.26 = 0.98560 / day
r = a(1 − ecos E )
Example Problems 1.1 Elliptic (Keplerian) Motion
Solution
Let r = distance between the Sun and the Earth a = semi-major axis of the Earth’s orbit around the Sun
Given: a = 1 AU (mean distance of the Sun from the Earth) e = 0.0167; n (mean motion) = 3600/365.25 0/day = 0.9865 0/day
Criteria for Earth to be farther from the Sun than 1 AU: r > a
Kepler’s Equation: M = E - e sin ERelationship of r to E: r = a (1- e cos E)
Therefore: r = a (1- e cos E) > aor cos E < 0 gives 900 < E < 2700
Substitute into Kepler’s Equation and solve for values of M:
M1 = 89.040 and M2 = 270.040
∆T = (M2 – M1) / n = (270.040 – 89.040) / 0.9865 = 184.6 days (The time that the Earth is farther away from the Sun than 1 AU)
Orbital Dynamics:General perturbation,Kaula’s formulation
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
Under the influence of the 'realistic' Earth (and ignoring other forces), the point mass potentialused to derive the elliptic motion have to be replaced by adding perturbing potential as follows:
(26) where and
Even though R is small compared to , it makes the orbital elliptic plane (its size, shape, andorientation) significantly change in time. Consequently, the Keplerian elements are time-varying,i.e.,
V∇=r&&
Rr
rV +=µ
λθ ),,(
( )[ ]λλφµ
λφ mSmCPr
a
rrRR lmlmlm
l
l
l
m
e sincossin ),,(2 0
+
== ∑∑
∞
= =
rµ
)(),(),(),(),( tttiiteetaa ωω =Ω=Ω===
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
(27)
(28) where i = 1, 2, and 3.
Def'n: osculating ellipse– It is an instantaneous ellipse defined by the corresponding Keplerian elements at a particularepoch. On the osculating ellipse, we will have the following constraints:
(29)
(30)
dt
ds
s
x
t
x
dt
dxx k
k
iiii ∂
∂+
∂
∂==&
dt
ds
s
x
t
x
dt
xdx k
k
iiii ∂
∂+
∂
∂==
&&&&&
Msssisesas =Ω===== 654321 , , , , , ω
t
x
dt
dx ii
∂
∂=
∂∂
=∂
∂
rxt
xi µ&
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
It leads to
(31)
(32) Equations (31) and (32) can be re-written as follows:
(33)
(34) where
Therefore,
(35)
0=∂
∂
dt
ds
s
x k
k
i
i
k
k
i
xR
dt
ds
s
x
∂∂
=∂
∂&
0sA =&
[ ]
RxR
i
∇=
∂∂
=×13
sA&&
[ ] [ ]6363
,××
∂
∂≡
∂
∂≡
k
i
k
i
s
x
s
x &&AA
[ ] [ ]TT , MÙùieaMÙùiea &&&&&&& == ss
[ ][ ]16
TT
×
∂
∂=⋅−
ks
RsAAAA &&&
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
The 6 by 6 square matrix, , is called Lagrange bracket. It is a skew-symmetric andtime-invariant matrix. Try to prove this. Note that the time-invariance permits to evaluate theLagrange bracket at any epoch (e.g., perigee passage time) and use it for the remaining epochswithout evaluating it every time.
AAAA TT && −
Remind
and are vectors indicating the location and motion of the satellite in the orbital ellipse(osculating ellipse) and the rotation matrix, R, orients them to the inertial frame.
( ) ( ) ( )MeaiMiea ,,,,,,,,, rRx ωω Ω=Ω
( ) ( ) ( )MeaiMiea ,,,,,,,,, rRx && ωω Ω=Ω
r r&
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
To compute components of the Lagrange bracket matrix, we consider the following:
(36)
(37)
Each partial can be computed using the followings:
(38)
(39)
rRRRrrr
R
rRrRx
∂∂
+∂∂
+∂∂
+
∂∂
+∂∂
+∂∂
=
⋅+⋅=
ÙÙ
ddii
ddMM
dee
daa
ddd
ωω
rRRRrrr
R
rRrRx
&&&&
&&&
∂∂
+∂∂
+∂∂
+
∂∂
+∂∂
+∂∂
=
⋅+⋅=
ÙÙ
ddii
ddMM
dee
daa
ddd
ωω
−−
+−
=∂∂
0
cos23
sin1
sin23
cos
2 EMra
Ee
EMra
eE
ar
( )
−−
+−
=∂∂
0
cos1
sin
sin1
2
2
eEe
Era
Era
aer
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
(40)
(41)
(42)
(43)
−
−
=∂∂
0
cos1
sin2
2
Ee
E
ra
Mr
( )
−−
−+
=∂∂
0
cossin3
1
cos3
sin
2 2
22
2
2
EEMra
e
eEMra
E
rna
ar&
( )
+−
−−−
+−
−=∂∂
011
coscos11
coscossin
22
2
3
ra
eeEEe
Ear
eEEra
rna
er&
−
−
−=∂∂
0
sin1
cos2
3
4
Ee
eE
rna
Mr&
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
(44)
(45)
(46)
−
Ω−Ω−Ω−
ΩΩΩ
=∂∂
iii
iii
iii
isincoscoscossin
coscossincoscossinsincos
cossinsincossinsinsinsin
ωω
ωω
ωωR
ΩΩ−Ω−Ω−Ω
ΩΩ−ΩΩ−Ω−
=Ω∂
∂
000
sinsincoscossinsincoscossinsincoscos
sincoscoscoscossinsincossincoscossin
iii
iii
ωωωω
ωωωωR
−
Ω−Ω−Ω+Ω−
Ω+Ω−Ω−Ω−
=∂
∂
0sinsinsincos
0cossincoscossincoscoscossinsin
0cossinsincoscoscoscossinsincos
ii
ii
ii
ωω
ωωωω
ωωωω
ωR
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
Each element in (36) and (37) can be computed using above equations. However, we know thereare only six non-zero elements in the Lagrange bracket matrix. The complete set of nonzeroelements is
(47)
(48)
(49)
(50)
(51)
(52)
[ ] [ ] ( ) ienai i sin1,2/122
),(TT −−=−=Ω ΩAAAA &&
[ ] [ ] ( ) iena
a a cos12
,2/12
),(TT −=−=Ω ΩAAAA &&
[ ] [ ]( ) ( ) 2/12
2
,TT
1
cos,
e
ienae e
−
−=−=Ω ΩAAAA &&
[ ] [ ]( ) ( ) 2/12,
TT 12
, ena
a a −=−= ωω AAAA &&
[ ] [ ]( ) ( ) 2/12
2
,TT
1,
e
enae e
−
−=−= ωω AAAA &&
[ ] [ ]( ) 2, ,
TT naMa e
−=−= ωAAAA &&
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
After substituting each partial to equation (35), the solution of the time-derivatives of six orbitalelements follows:
(53)
(54)
(55)
(56)
(57)
(58)
This set of equations is called Lagrange Planetary Equation (LPE). The "weak" singularityappearing the equations can be easily removed by introducing Delaunay elements. For detaildescription, we refer Kaula (1966), pp.29-30.
M
R
nadt
da
∂
∂=2
ω∂∂−
−∂
∂−=
R
ena
e
M
R
ena
e
dt
de2
2
2
2 11
a
R
nae
R
ena
en
dt
dM
∂
∂−
∂
∂−−=
212
2
Ω∂
∂
−−
∂
∂
−=
R
iena
R
iena
i
dt
di
sin1
1
sin1
cos2222 ω
i
R
ienadt
d
∂
∂
−=
Ω
sin1
122
e
R
ena
e
i
R
iena
i
dt
d
∂
∂−+
∂
∂
−−=
2
2
22
1
sin1
cosω
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
In LPE, we still have partials need to be specified. The perturbing potential, R, given in equation(26) is expressed in terms of the Earth fixed spherical coordinates, . Here, we will try toexpress the perturbing potential, R, using the orbital elements by investigating the relationshipbetween the orbital elements and the Earth fixed spherical coordinates.
Figure 3 shows the longitude relationship:. (59)
α – right ascensionGST – Greenwich Sidereal Time
( )λθ ,,r
equinox
Greenwich
node
satellite
Ω
α
GST λ
( ) ( )GST−Ω+Ω−= αλ
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
The spherical trigonometry indicates the latitudinal relationship as follows:
(60)
(61)
(62)
equinox
node
satellite
Ωνω +
Ω−αi
φ
( ) ( )φνω
αcos
coscos
+=Ω−
( ) ( )φνω
αcos
cossinsin
i+=Ω−
( ) isincossin νωφ +=
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
Equations (59) to (62) show the relationship between the Earth fixed latitude and longitude to theorbital elements. Finally, we have the following equation for the radius:
(63)
(64)
where is the eccentricity function.
After elaborated mathematical manipulation (Kaula, 1966), we finally obtain the expression ofresidual potential in terms of orbital elements as follows:
(65)
where is the inclination function.
( )( ) ( )( )( )( ) ( )( )
( )
=
−Ω++−
−Ω++−∑∞
−∞=++
lmpq
lmpq
qlpqlleG
aGSTmpl
GSTmpl
r ψ
ψ
νω
νωsin
cos12sin
2cos111
( ) ( ) ( )GSTmMqplpllmpq −Ω++−+−= 22 ωψ
( )eGlpq
( )
∑ ∑∑ ∑∞
= = = −=
−
−
+−
+
=
Ω=
2 0 0
1
1
even :
odd : sincos
sincos )()(
,,,,,
l
l
m
l
p q
ml
mllmpqlmlmpqlm
lmpqlmlmpqlmlpqlmp
l
e
CS
SCeGiF
a
a
a
MieaRR
ψψ
ψψµ
ω
)(iFlmp
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
(Exercise)At this moment, it is worth to exercise computing the frequency-independent orbital perturbationinduced by the second zonal coefficient of the residual potential. These perturbations can be usedas reference elements, because the second zonal coefficient exceeds other coefficients'contribution by a factor of 1000. The frequency, , is determined by indices, l, m, p, and q. Consequently, the frequencyindependence yields and comes from the condition of
From this, we immediately identify the even zonal harmonics provide the frequency independentperturbation. For the case of (l=2, m=0, p=1, and q=0), we can find the corresponding inclinationand eccentricity functions from the appropriate equations (or just from the tables given in Kaula,1966).
lmpqψ&
0 and 0, ,02 ===− mqpl
21sin43)( 21,0,2 −= iiF
( ) 2/320,1,2 1)( eeG −=
Notes from S. Han
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
The corresponding residual potential follows:
By putting it to LPE, we get
We identify the constant a, e, and i and consequently, constant change of Ω, ω, and M. Theyindicate fixed size and shape of the orbital ellipse, fixed inclination, precessing node and perigee.From these equations, we can interpret the perigee will retrograde if the inclination is bounded by63.4° and 116.6°, because the second zonal coefficient is negative. Also, the node will retrogradefor the inclinations smaller than 90°. [See example problem next]
( ) ( ) 2022/32
3
2
20 21sin431 Ciea
aR
e−−=
−µ
( )
( )( )
( )( )1cos3
14
3
cos5114
3
cos12
3
0
0
0
2
22/32
220
2
222
220
222
220
−−
−=
−−
=
−=
Ω
=
=
=
iae
aCnn
dt
dM
iae
aCn
dt
d
iae
aCn
dt
d
dt
didt
dedt
da
e
e
e
ω
Notes from S. Han
Example Problem: J2 Perturbations
A scientific geodetic satellite was launched evolving an unknown planet in a circular orbit, with aninclination of 300 and a semi-major axis of 6,500 km. The mean radius and µ of the planet areassumed known at 6,000 km and 3.986x1014 m3/s2, respectively. The ascending node of theorbit is observed to be moving from west to east (progressing) and is completing a revolutionin 40 earth-days. Assuming Keplerian motion and J2 is only perturbation on the satellite,
(a) Determine J2 of the planet. Give a physical description of the shape of the planet andcontrast the property with that of the Earth.
(b) Calculate ˙ ω . Explain the meaning of this quantity.(c) If the orbit is desired to be frozen (i.e., ˙ ω = 0), what is the orbital inclination, i?(d) State the range of the inclination values of the satellite orbit for the cases:
˙ ω < 0 , ˙ ω > 0˙ Ω < 0 , ˙ Ω > 0
Example Problem Sollution: J2 Perturbations
€
1.e = 0 (Circular orbit)i = 30°,a = 6500kmR = 6000km
µ = 3.986×1014 m3
s2
Ω⋅
=2 ×π
40 × 24 × 60× 60sec=1.81805×10−6
n =µa3 =1.20475 ×10−3
Substituting for the given values and solving for J2
Ω⋅
= −32nJ2
Ra
2 cosi1− e2( )2
J2 = −1.36335×10−3
Since J2 < 0⇒ the shape of the planet is oblongJ2 > 0⇒ the shape of the planet is oblate (in the case of Earth
Example Problem Sollution: J2 Perturbations
€
2.
ω⋅
= −34nJ2
Ra
2 1− 5cos2 i( )1− e2( )2
= −34nJ2
Ra
2
1− 5cos2 i( ) (as e = 0)
⇒ω⋅
= −14.28942°day
€
3. ω⋅
= 0⇒1− 5cos2 i = 0
⇒ cosi =15
⇒ i = 63.43°or i =116.57° Critical inclination
Example Problem Solution: J2 Perturbations
€
4. For ω⋅
> 0⇒ 5cos2 i −1> 0
⇒ cosi − 15
cosi + 1
5
> 0
⇒0 ≤ i < 63.43494° or 116.56505° < i ≤180°
if ω⋅
< 0⇒ 5cos2 i −1< 0
⇒ cosi − 15
cosi + 1
5
< 0
⇒ 63.43494°<i < 116.56505°
if Ω⋅
> 0⇒ cosi < 0⇒ 90 < i <180
if Ω⋅
< 0⇒ cosi > 0⇒ 0 < i < 90
Orbital Dynamics:Periodic variations andresonances due togeopotential
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
Using Kaula’s expression for perturbing potential ( Ulmpq , as a f unction of Keplerian orbit
elements) and employing the Lagrangian planetary equations, derive ∆ω and ∆Ω. Steps: integratethe Lagrangian planetary equation, and express ∆Ω as a function of F(i), G(e), and S:
€
dΩdt
=1
n a2 1− e2 sini∂R∂i
;
€
Ulmpq =µae
l
al +1 Flmp (i)Glpq (e)Slmpq (ω,M,Ω,θ)
where
[ ]
[ ])()2()2(sin
)()2()2(cos
θω
θω
−Ω++−+−
+
−Ω++−+−
−=
−
−
−
−
mMqplplC
S
mMqplplS
CS
evenml
oddmllm
lm
evenml
oddmllm
lmlmpq
If we assume that the only variations with time of the elements are θω &&&& ,,, MΩ , the argument ofthe trigonometric function can be expressed as follows:
)()()(
)()2()2()(
000 tttt
mMqplplt
ψψ
θωψ
&−+=
−Ω++−+−=
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
ωΔ
),,,()(
)(1
)()(
sin1
cos
1
sin1
cos
1
sin1
cos
12
2
122
2
2
22
2
2
22
θωµµ
ω
Ω
∂
∂−+
∂
∂
−−=
∂
∂−+
∂
∂
−−=
∂∂−
+∂∂
−−=
++ MSe
eGiF
aa
enae
eGi
iF
aa
iena
i
e
U
enae
i
U
iena
i
eR
enae
iR
iena
idt
d
lmpqlpq
lmpl
le
lpqlmp
l
le
lmpqlmpq
lmpq
Also,
€
Slmpqt0
t∫ dt =
Clm
−Slm
l−m odd
l−m even
cos(ψ(t)t0
t∫ dt +
Slm
Clm
l−m odd
l−m even
sin(ψ(t)t0
t∫ dt
=Clm
−Slm
l−m odd
l−m even sin ψ(t)( ) − sin ψ(t0)( )˙ ψ (t0)
−Slm
Clm
l−m odd
l−m even cos ψ(t)( ) − cos ψ(t0)( )˙ ψ (t0)
=1
˙ ψ (t0)Clm
−Slm
l−m odd
l−m even
sin ψ(t)( ) −Slm
Clm
l−m odd
l−m even
cos ψ(t)( )
≡ S lmpq
1 2 4 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
€
+1
˙ ψ (t0)−Clm sin ψ(t0)( ) + Slm cos ψ(t0)( )Slm sin ψ(t0)( ) + Clm cos ψ(t0)( )
l−m odd
l−m even
= 0 (Qsince the total perturbation at time t 0 is zero)1 2 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4
=1
˙ ψ (t0)S lmpq
After integration,
( ) ( ) ( )[ ][ ])()2()2(
)(/1cot/)(1
)(1)(
)(1
)()(
sin1
cos
3
2/1212
012
2
122
0
θωµ
ψµµ
ωω
&&&&
&
−Ω++−+−
∂∂−−∂∂−=
∂
∂−+
∂
∂
−−=
=Δ
+
−−
++
∫
mMqplplna
SeGiFeieGiFeea
Ste
eGiF
aa
enae
eGi
iF
aa
iena
i
dtdt
d
llmpqlpqlmplpqlmpl
e
lmpqlpq
lmpl
le
lpqlmp
l
le
t
t
lmpqlmpq
∑Δ=Δnmpq
lmpqωω
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
ΔΩ
),,,()()(
sin1
1
sin1
1
sin1
1
122
22
22
θωµ
Ω
∂
∂
−=
∂
∂
−=
∂∂
−=
Ω
+ MSeGi
iF
aa
iena
i
U
iena
iR
ienadt
d
lmpqlpqlmp
l
le
lmpq
lmpq
Similarly,
( )[ ])()2()2(sin1
)(/
)(1
)()(
sin1
1
23
0122
0
θωµ
ψµ
&&&&
&
−Ω++−+−−
∂∂=
∂
∂
−=
Ω=ΔΩ
+
+
∫
mMqplpliena
SeGiFa
St
eGi
iF
aa
iena
dtdt
d
l
lmpqlpqlmple
lmpqlpqlmp
l
le
t
t
lmpqlmpq
∑ΔΩ=ΔΩnmpq
lmpq
Similarly for other orbit elements
1. Dynamics of Satellite Orbits 1.2. Perturbed Motion
The frequency of the perturbation can be rearranged as follows (Schrama, 1989):
(66)
Form this representation, we identify three major frequencies. The first one is once-per-revolution component with and l-2p+q controls how many cycles per revolution. Thesecond one is daily component with and m controls how many cycles per day. Thethird one is long period component with . Note that many combinations of such indices canproduce the same frequency. In case of l=2p and q=0, the frequency is determined solely by thespherical harmonic order, m. producing m-daily components. The perturbation induced by aspecific order and all even degrees overlap on the same frequency (m cycles per day). Putting itanother way, the satellite with m cycles per day is sensitive to the resonant effect from the orderm and all even degree geopotential coefficients.
( )( ) ( ) ωωψ &&&&&& qTSGmMqpllmpq −−Ω+++−= 2
M&& +ωTSG && −Ω
ω&
Notes from S. Han
Example Problem: Resonances for GRACE
Find the primary and secondary resonance and the respective resonance periods (in days) for thesatellite GRACE, assuming an orbital altitude of 450 km, i = 89°.
€
˙ ω = −4.035° /day; ˙ Ω = −0.141° /day; ˙ M ≅ 5000° /day; ˙ θ ≅ 360° /day˙ ψ lmpq = (l − 2p) ˙ ω + (l − 2p + q) ˙ M + m( ˙ Ω − ˙ θ )
The resonance occurs when 0)()2()2( ≈−Ω++−+−= θωψ &&&&& mMqplpllmpq .
Assume 0=q , then
( ) ( ) ( )1487.13
360141.05000035.4
2
022
≈=−−+−
−=−Ω
+−=
−
≈−Ω+−+−
θω
θω
&&
&&
&&&&
Mpl
m
mMplpl
Example Problem: Resonances for GRACE (continued:)
For the primary resonance .12 ≈− plThen, the primary resonance occurs at .14=m
( )day
daydaydaydaymMlmpq
/009.46
)/360/141.0(14/5000/035.4o
oooo&&&&&
−=
−−×++−=−Ω++= θωψ
Primary resonance period = days82.72
=ψπ&
For the secondary resonance .22 ≈− pl
Then, .28=m( )
( )day
daydaydayday
Mlmpq
/012.92
)/360/141.0(28/50002/035.42
2822
o
oooo
&&&&&
=
−−×+×+−×=
−Ω×+×+×= θωψ
Secondary resonance period = days91.32
=ψπ&
Example Problem: Orbital variations due to geopotential
Determine all the frequencies (expressed in terms of angular rates, e.g., ˙ ω , M , ˙ Ω , et c) of
€
˙ ψ lmpqassociated with C30. Justify the ranges of the indices (l, m, p, q). Categorize frequencies, ifavailable, which are associated with secular, short period, long period, m-daily, and resonances.
Example Problem: Orbital vairations due to geopotential (Continued:)