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

qq

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: jklokocn@asu.cas.cz

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)