CHAMP’s TRIPLE PASSAGE THROUGH 31 st /62 nd -ORDER ORBIT RESONANCE
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Transcript of CHAMP’s TRIPLE PASSAGE THROUGH 31 st /62 nd -ORDER ORBIT RESONANCE
CHAMP’s TRIPLE PASSAGE THROUGH 31st/62nd-ORDER
ORBIT RESONANCE
R.H. Gooding, C.A. Wagner,
J. Klokočník, J. Kostelecký
Lagrange planetary equationsthe case of orbital inclination
1. Allan/Kaula expression2.after choice of resonant indices
3.final resonant form with lumped coefficients
1.
Lagrange Planetary Equations with resonant choice for (l,m,p,q)
2
10/
eOLCeLCdtdI
Lagrange Planetary Equation for Orbital Inclinationin terms of Lumped Geopotential Coefficients (LC)
DEFINITIONSResonant angle
Lumped coefficients
Location of resonances
Location of resonances
what is semi-major axis / mean motion
at exact resonance
Two types of semi-major axis:
Brouwer
Kozai
what is semi-major axis / mean motion
at exact resonance
Two types of semi-major axis:
Brouwer
Kozai
CHAMP and RESONANCES
• CHAMP mission • Location of resonances in CHAMP orbit• simulation of forthcoming resonances in
inclination• preparation for analysis of individual
resonances• analysis of inclination variations at 46/3 and
31/2 resonances
Comparison of approaches to analyse CHAMP resonances: long-arc vs short-arc
• general geopotential recovery from tracking data is to analyse full spectrum of effects in many short-arcs
[e.g. 1.5 day for CHAMP]
• traditional “resonant analyses” work with long-arc approach and concentrate on few “resonant frequencies” (31/2, 46/3….etc)
CHAMP - inclination
87.245
87.25
87.255
87.26
87.265
87.27
87.275
87.28
51600 51800 52000 52200 52400 52600 52800 53000
MJD
de
gre
e
CHAMP - semimajor axis
6760
6770
6780
6790
6800
6810
6820
6830
6840
51600 51800 52000 52200 52400 52600 52800
MJD
km
CHAMP - eccentricity
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
0.005
51600 51800 52000 52200 52400 52600 52800 53000
MJD
CHAMP inclination variations due to different resonances (GRIM5-C1, 90 x 90)
-4
-2
0
2
4
6
8
10
51600 51800 52000 52200 52400 52600 52800
time [MJD]
chan
ge
of
incl
inat
ion
[1e
-4 d
eg]
res. 46/3
res. 77/5
res. 31/2
46/3 (4 Oct. 00) 77/5 (23 Sep. 01) 31/2 (24 May 02)
orbital maneouvre 11 Jun. 02
31/2 (28 Oct. 02)
orbital maneouvre 9 Dec. 02
CHAMP inclination variations due to three different resonances (Status July 2003)
-4
-2
0
2
4
6
8
10
12
51600 51800 52000 52200 52400 52600 52800 53000
MJD
incl
inat
ion
[1d
-4 d
eg
]
46/3 (4.10.2000)77/5 (23.9.2001)
31/2 (25.5.2002)
31/2 (30.10.2002)
31/2 (11.6.03)
orbit manoeuvre 11.6.2002
orbit manoeuvre 9.12.2002
-4
-2
0
2
4
6
8
10
12
51600 51800 52000 52200 52400 52600 52800 53000
time [MJD]
incl
inat
ion
[1d
-4 d
eg
]
CHAMP inclination variations due to three different resonances (Status August 2003)
46/3 (4.10.2000) 77/5 (23.9.2001)
31/2 (25.5.2002)
31/2 (30.10.2002) 31/2 (11.6.03)
orbit manoeuvre 11.6.2002
orbit manoeuvre 9.12.2002
Resonance 31/2
CHAMP - res. 31/2 - model
-8
-6
-4
-2
0
2
4
6
8
10
12
52200 52300 52400 52500 52600 52700 52800 52900 53000 53100
MJD [days]
incl
inat
ion
[1d
-4 d
eg
]
GRIM-5C1
EIGEN-3P
Resonant angle of three 31/2 orbital resonance of CHAMP
0
50
100
150
200
250
300
350
400
52350 52400 52450 52500 52550 52600 52650 52700 52750 52800 52850 52900
MJD
de
gre
e
31/2 (25.5.02)
31/2 (30.10.02)
31/2 (11.6.03)
31:2 18-para fit, I for 2 arcs
87.2609
87.2611
87.2613
87.2615
87.2617
87.2619
87.2621
0 50 100 150 200 250 300
Days from MJD 52300
Incl
inat
ion
Champ's 31:2 Resonance
87.2609
87.2612
87.2615
87.2618
87.2621
87.2624
87.2627
1 51 101 151 201 251 301 351 401 451 501 551
Days from MJD 52299
Inc
lina
tio
n
Lumped coefficients C(31,0,2)
-8
-6
-4
-2
0
2
4
6
8
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140
inclination
Fc
x 10
e9 GRIM5C1
Lumped coefficient S(31,0,2)
-4
-3
-2
-1
0
1
2
3
4
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140
inclination
Fs
x 10
e9 GRIM5C1
Lumped coefficient C(31,0,2) - model EIGEN-1S
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140
inclination [deg]
Fc x
10e
9
Lumped coefficient S(31,0,2) - model EIGEN-1S
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140
inclination [deg]
Fs x
10e
9
Lumped coefficient C(31,0,2) - model EIGEN-2
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140
inclination [deg]
Fc x
10e
9
Lumped coefficient S(31,0,2) - model EIGEN-2
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140
inclination [deg]
Fs x
10e
9
FC - CHAMP - 31/2
0.8
0.9
1.0
1.1
87.2 87.25 87.3 87.35 87.4
inclination [deg]
FC x
1e
9
GRIM5-C1
PGM2000
EGM96
EIGEN-1S
EIGEN-2
minus
plus
EIGEN-3p
minus
plus
resonant TLE CAW 2/04
FS - CHAMP - 31/2
0.3
0.4
0.5
0.6
0.7
0.8
87.2 87.25 87.3 87.35 87.4
inclination [deg]
FS x
1e9
GRIM5-C1
PGM2000
EGM96
EIGEN-1S
EIGEN-2
minus
plus
EIGEN-3p
minus
plus
resonant TLE CAW 2/04
FC - CHAMP - 31/2
0.8
0.9
1.0
1.1
87.2 87.22 87.24 87.26 87.28 87.3 87.32 87.34 87.36 87.38 87.4
inclination [deg]
FC
x 1
e9
GRIM5-C1
PGM2000A
EGM96
EIGEN-1S
EIGEN-2
EIGEN-3P
resonant TLE
resonant POME
FS - CHAMP - 31/2
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
87.2 87.22 87.24 87.26 87.28 87.3 87.32 87.34 87.36 87.38 87.4
inclination [deg]
FS
x 1
e9
GRIM5-C1
PGM2000A
EGM96
EIGEN-1S
EIGEN-2
EIGEN-3P
resonant TLE
resonant POME
FC - CHAMP - 62/4
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
87.2 87.22 87.24 87.26 87.28 87.3 87.32 87.34 87.36 87.38 87.4
inclination [deg]
FC x
1e
9
GRIM5-C1
PGM2000A
EGM96
EIGEN-1S
EIGEN-2
EIGEN-3P
resonant TLE
resonant POME
FS - CHAMP - 62/4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
87.2 87.22 87.24 87.26 87.28 87.3 87.32 87.34 87.36 87.38 87.4
inclination [deg]
FS x
1e
9
GRIM5-C1
PGM2000A
EGM96
EIGEN-1S
EIGEN-2
EIGEN-3P
resonant TLE
resonant POME
Conclusions
• Introduction to theory of resonant phenomenon in orbits of Earth artificial satellites
• Historical analyses (Gooding etc)• lumped coefficients• rotation of upper atmosphere• calibration of comprehensive
solutions for geopotential
CHAMP:
•Location, estimation of expected orbit effects and analysis of particular high-order resonances in CHAMP orbit
•Comparison of computed lumped geopotential coefficients from resonances with those from comprehensive Earth models
•Interpretation of existing discrepancies (due mainly to insufficient modelling of tides and non- gravitational effects in our resonant software), and a possibility to calibrate the Earth models by results from resonances
To get a copy
• Anonymous ftp: sunkl.asu.cas.cz
• pub/jklokocn files: PPT_RES_CHAMP_GFZ.ppt
• PPT_RESON.ppt
• Web: www.asu.cas.cz/~jklokocn
• e-mail: [email protected]
The End
Variation of inclination of CHAMP due to 31/2 resonances in 2002 and 2003
87.2618
87.2620
87.2622
87.2624
87.2626
87.2628
87.2630
87.2632
87.2634
52200 52300 52400 52500 52600 52700 52800 52900
MJD
incl
inat
ion
[d
eg
]
31/2 (25.5.2002) 31/2 (30.10.2002) 31/2 (11.6.03)