2. RESONANCES AND GRAVITY FIELD OF THE EARTH J. Klokočník, J. Kostelecký LECTURE AT NRIAG HELWAN EGYPT November 2005

4. NASA about 1970 unexplained orbit oscillations RAE Farnborough, pioneer works of R.H.Gooding and R.R.Allan, case of Ariel 3 at the 15 th -order resonance RAE Farnborough, group of D.G. King-Hele, many analysis of inclination and eccentricity for orbits at 15/1, 14/1, 29/2, and 31/2 resonances for diverse inclinations, application in gravity field models accuracy tests for the particular orders of the harmonic geopotential coefficients USA NASA GSFC, later NOAA Carl Wagner, Steve Klosko, mostly shallow resonance France and Germany: G. Balmino and Ch. Reigber an their groups, shallow resonances Czechoslovakia, Czech republic: Klokocnik and Kostelecky, Ondrejov Observatory

7. Lagrange Planetary Equation for Orbital Inclination in terms of Lumped Geopotential Coefficients (LC)

8. DEFINITIONS Resonant angle Lumped coefficients

11. CHAMP, GRACE A, B and RESONANCES comments on theory location of resonances in their orbits simulation of resonances in inclination preparation for analysis of individual resonances analysis of inclination variations at 46/3 and 31/2 resonances of CHAMP, preparation for 61/4 of GRACE

12. 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)

15. CHAMP GFZ 2000

17. numbers on the figure are the number  of nodal revolutions per of  synodic days

27. Eigen 3p- generated

28. numbers on the figure are the number  of nodal revolutions per of  synodic days

30. DEFINITIONS Resonant angle Lumped coefficients

31. Data NASA or GFZ two line elements – mean Kozai type elements POME – state vectors for CHAMP and GRACE A/B, transformation from CTS to TDS and the to osculating elements, subsequent averaging to mean elements, subtracting all non-resonant perturbations (lunisolar, tidal, due to rotation of the upper atmosphere) or additional empirical parameters in the LSE

32. Resonance 46/3

40. Resonance 31/2

41. numbers on the figure are the number  of nodal revolutions per of  synodic days

47. Eigen 3p- generated

48. first exact. Res.: 23 May 02, day 143

70. 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

71. Values of the best fitting lumped coefficients from analysis of POME Gooding, THROE CAW 31st order: 10 9 C(31, 0, 2) = -16.47 ± 0.20 -16.27 ± 0.14 10 9 S(31, 0, 2) = - 8.35 ± 0.19 - 9.03 ± 0.15 62nd order: 10 9 C(62, 0, 4) = 2.62 ± 0.08 2.79 ± 0.14 10 9 S(62, 0, 4) = 0.41 ± 0.08 0.38 ± 0.05

72. Champ 78/5 resonance

74. Champ 47/3 resonance will occur in winter 2005/6

75. Expected 47/3 resonance in orbit of CHAMP under different assumptions on atmospheric drag Expected 47/3 resonance in orbit of CHAMP under different assumptions on atmospheric drag

78. Grace A and B, 2002

80. GRACE at 61/4 is the next

84. Resonant inclination changes Eigen 3p-generated

91. Bezdek 2005

93. Conclusions CHAMP orbit was raised twice and so CHAMP passed through the 31/2 resonance three times – this gives unprecedented opportunity for determining the values of lumped coefficients of orders 31 and 62, via recently revived technique that capitalizes on the resonant variation of orbit inclination (in particular) Resonant analysis is based on TLE and POME of CHAMP Lumped coefficients of the 31 st and 62 nd order compare well with relevant values computed from recent comprehensive gravity field models (like EIGEN 2, 3p etc). Resonant analysis still useful to check/calibrate selected gravity information. Possibilities and capabilities to improve the procedure…doors open for Yehia AbdelAziz and others Forthcoming: 47/3 resonance of CHAMP (winter 2005/6), study of 61/4, 46/3, 77/5, 31/2, 47/3 of GRACE Unexpected applications in solving a problem of accuracy of determination of geopotential time variations (case of 61/4 of GRACE A/B) in 2004

94. To get a copy Anonymous ftp: sunkl.asu.cas.cz pub/jklokocn file: PPT_RESON_EGYPT05.ppt For more information: Web: www.asu.cas.cz/~jklokocn e-mail: jklokocn@asu.cas.czjklokocn@asu.cas.cz kost@fsv.cvut.cz

95. The End