satellite geodesy: kepler orbits, kaula ch. 2+3.i1.2a
DESCRIPTION
Satellite geodesy: Kepler orbits, Kaula Ch. 2+3.I1.2a. Basic equation: Acceleration Connects potential, V, and geometry. (We disregard disturbing forces – friction). C.C.Tscherning, 2007-10-25. Velocity: Integration along orbit. Position: one integration more. C.C.Tscherning, 2007-10-25. - PowerPoint PPT PresentationTRANSCRIPT
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Satellite geodesy: Kepler orbits, Kaula Ch. 2+3.I1.2a
• Basic equation: Acceleration
• Connects potential, V, and geometry. (We disregard disturbing forces – friction).
C.C.Tscherning, 2007-10-25.
i
ii
i
X
V
dt
tXdtX
tX
2
2 )()(
timeoffunction a as scoordinate-orbit)(
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Velocity: Integration along orbit.
• Position: one integration more
C.C.Tscherning, 2007-10-25.
)()()(
)( 02
2
1
1
0
1tXdt
dt
tXd
dt
tdXtX i
t
t
it
ii
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
• We must know (approximatively) the orbit to make the integration.
• The equation connects the position and velocity with parameters expressing V.
• Parameters: kMCij
Orbit integration and parameter determination
C.C.Tscherning, 2007-10-25.
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
• Directions and distances from Earth using Cameras, lasers, radar-tracking, time-differences
• Distances from satellites to ”point” on Earth surface (also ”cross-overs”)
• Range rates: Doppler effect, contineous tracking.
• Measurements in or between satellites: gradiometry, GPS-positions, ranging
Observations
C.C.Tscherning, 2007-10-25.
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
• Spherical harmonic coefficients, kMCij
• Positions of ground tracking stations
• Changes to Earth Rotation and pole-position
• Tides (both oceanic and solid earth)
• Drag-coefficients, air-density
• Contributions from Sun and Moon.
Parameters
C.C.Tscherning, 2007-10-25.
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Ordinary differential equations
• Change from 3 second order equations to 6 first-order equations:
C.C.Tscherning, 2007-10-25.
i
i
i
i
i
X
V
dt
dXdt
X
X
V
dt
X
dY:1'.order of 6
d :order 2'. of 3
i
2
2
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Coordinate transformation in 6D-space
• New coordinates qi and pi .
C.C.Tscherning, 2007-10-25.
),,,,,(
),,,,,(
321321
321321
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XXXXXXV
q
V
dt
dp
p
V
dt
dq
i
i
i
ii
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
(q,p) selected so orbits straight lines
• If
• Possible also so that kmCij ”amplified”.
C.C.Tscherning, 2007-10-25.
0 ,0
p)(q, find topossible ,
dt
dp
dt
dqr
kMV
ii
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Kepler orbit
• If potential V=km/r:• Orbit in plane through origin (0).• Is an ellipse with one focus in origin
C.C.Tscherning, 2007-10-25.
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Geometry
• E and f
C.C.Tscherning, 2007-10-25.
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Kepler elements
• i=inclination, Ω=longitude of ascending node (DK: knude)
• e=excentricity, a=semi-major axis,
• ω=argument of perigaeum, f+ ω=”latitude”.
• M=E-esinE=Mean anomaly (linear in time !)
C.C.Tscherning, 2007-10-25.
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
From CIS to CTS
• We must transform from Conventional Inertial System to Conventional Terrestrial System using siderial time, θ:
• Rotation Matrix
C.C.Tscherning, 2007-10-25.
3
2
1
3
100
0cossin
0sincos
)(
u
u
u
uR
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
From q-system to CIS
• 3 rotations. Ri with integer i subscript is rotation about i-axis. Rxu is rotation from u to x.
xuqxquqx RRRRiRRR ),()()( 313
C.C.Tscherning, 2007-10-25.
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Elliptic orbit
• We use spherical coordinates r,λ in (q1,q2)-plane
vector-unit is // 3 rxrxkx
C.C.Tscherning, 2007-10-25.
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Angular momentum
• λ is arbitrary := 0 !
C.C.Tscherning, 2007-10-25.
xyyxx
xyyx
xy
xy
hr
rr
rrr
h and /1
1
so ),/(tan
! conserved momentumangular
:r with gmultiplyin ,02
/)(
222
1
2
22
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Integration
• With u=1/r
C.C.Tscherning, 2007-10-25.
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Integration
C.C.Tscherning, 2007-10-25.
20
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2
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/
/1
so ,)r(-rin result theUse
hAur
hud
ud
ruhud
uduh
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Ellipse as solution
• If ellipse with center in (0,0)
C.C.Tscherning, 2007-10-25.
)1(
)cos(
)1(
1
r
1or
))cos(1(
e-a(1r
degree-2. of polynomiumith equation w
),1( with),sin(
)cos(
r)(f, scoordinate spherical use weand ,1
22
2)
222
2
2
2
2
ea
fe
eafe
eabfr
feaeba
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Expressed in orbital plane
C.C.Tscherning, 2007-10-25.
)cos1(
,sin1
)(cos
:E anomaly, excentric using expresed
)1(,)1(
,
22
21
22
1
220
Eeaqqr
Eeaq
eEaaeq
eahea
eAf
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Parameter change
C.C.Tscherning, 2007-10-25.
)derivation long a(after 1/
)2/()cos12)sin(2(
then ,)1(
sin)1(
)/1(
:E with f substitute weand , then
22
221
22
212
2
2
22
2
eeq
rEeaqEaqdE
dr
qqrdfea
qerdr
fea
er
df
rdr
df
dr
hfrf
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Further substitution
C.C.Tscherning, 2007-10-25.
2/32/1
0
2
222
222
and perigaeum of passagefor me with t ti
),()cos1(
)1(
)1
)1()cos1(
an
ttnMEe
ea
dt
dEq
e
e
qer
eaEea
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Transformation to CIS
C.C.Tscherning, 2007-10-25.
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Velocity
•
C.C.Tscherning, 2007-10-25.
ar
vVT
22
2
UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
From orbital plane to CIS
• .
C.C.Tscherning, 2007-10-25.
f. findcan weifknown )/(tan
then,)()(
put we),/)((tan
))/((tan
)0,,( toorthogonal ),,(
121
21
32/12
22
11
211
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xRiRp
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UNICA EU Research Contracts Seminar, Brussels, 29-30 May 2000.05.23
Determination of f
• .
C.C.Tscherning, 2007-10-25.
))/(cossin1(tan)/(tan
))(/(sin
) (from )/()(cos
))/(1(),2/(
)/(
2112
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2/1222
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