1. On the geodetic rotation of the major planets, the Moon and the Sun G.I. Eroshkin and V.V. Pashkevich Central (Pulkovo) Astronomical Observatory of the Russian Academy of Science St.Petersburg Space Research Centre of the Polish Academy of Sciences Warszawa 2009

2. The problem of the geodetic (relativistic) rotation of the major planets the Moon and the Sun (Eroshkin G.I., Pashkevich V.V., 2007) is studied by using the DE404/LE404 ephemeris, with respect to the proper coordinate systems of the bodies (Seidelmann P.K. et al., 2005). For each body the files of the Euler angles of the geodetic rotation are determined over the time span from AD1000 to AD3000 with one day spacing. The most essential terms of the geodetic rotation are found by means of the least squares method and spectral analysis methods. The mean longitudes of the planets and the Moon adjusted to the DE404/LE404 ephemeris were taken from (Brumberg and Bretagnon, 2000). The mean longitudes of Pluto adjusted to the DE404/LE404 ephemeris was taken from previous investigation (Eroshkin G.I., Pashkevich V.V., 2007). Aim:

3. Here the indices i and j refer to major planets, the Moon and the Sun; G – gravitational constant; – mass of the j-th body; c – velocity of light in vacuum; – barycentric positions and velocities of these points; sign × stands for the vector product. The angular velocity vector of the geodetic rotation for any body of Solar system:

4. Fig.1. Reference system used to define orientation of the planet.

5. Table 1. Recommended values for the direction of the north pole of rotation and the prime meridian of the Sun and planets (2000) (Seidelmann et al., 2005) Mercuryα 0 =281°.01-0°.033T δ 0 =61°.45-0°.005T W=329°.548+6°.1385025d Saturnα 0 =40°.589-0°.036T δ 0 =83°.537-0°.004T W=38°.90+810°.7939024d Venusα 0 =272°.76 δ 0 =67°.16 W=160°.20-1°.4813688d Uranusα 0 =257°.311 δ 0 =-15°.175 W= 203°.81-501°.1600928d The Earth α 0 =0°.00-0°.641T δ 0 =90°.00-0°.557T W=190°.147+360°.9856235d Neptuneα 0 =299°.36+0°.70 sin N δ 0 =43°.46-0°.51 cos N W=253°.18+536°.3128492d -0°.48 sin N N=357°.85+52°.316T Mars α 0 =317°.68143-0°.1061T δ 0 =52°.88650-0°.0609T W=176°.630+350°.89198226d Plutoα 0 = 313°.02 δ 0 =9°.09 W=236°.77-56°.3623195d Jupiterα 0 =268°.05-0°.009T δ 0 =64°.49+0°.003T W=284°.95+870°.5366420d The Sunα 0 =286°.13 δ 0 =63°.87 W=84°.10+14°.1844000d d - interval in days from J2000; T - interval in Julian centuries (of 36525 days) from J2000.

6. Table 2. Recommended values for the direction of the north pole of rotation and the prime meridian of the Moon (2000) (Seidelmann et al., 2005) The Moon α 0 =269°.9949 +0°.0031T -3°.8787 sin E1 -0°.1204 sin E2 +0°.0700 sin E3 -0°.0172 sin E4 +0°.0072 sin E6 -0°.0052 sin E10 +0°.0043 sin E13 δ 0 = 66°.5392 +0°.0130T +1°.5419 cos E1 +0°.0239 cos E2 -0°.0278 cos E3 +0°.0068 cos E4 -0°.0029 cos E6 +0°.0009 cos E7 +0°.0008 cos E10 -0°.0009 cos E13 W=38°.3213 +13°.17635815d -1°.4 x 10 -12 d 2 +3°.5610 sin E1 +0°.1208 sin E2 -0°.0642 sin E3 +0°.0158 sin E4 +0°.0252 sin E5 -0°.0066 sin E6 -0°.0047 sin E7 -0°.0046 sin E8 +0°.0028 sin E9 +0°.0052 sin E10 +0°.0040 sin E11 +0°.0019 sin E12 -0°.0044 sin E13 E1=125°.045-0°.0529921d, E2=250°.089-0°.1059842d, E3=260°.008+13°.0120009d, E4=176°.625+13°.3407154d, E5=357°.529+0°.9856003d, E6=311°.589+26°.4057084d, E7=134°.963+13°.0649930d, E8=276°.617+0°.3287146d, E9=34°.226+1°.7484877d, E10=15°.134-0°.1589763d, E11=119°.743+0°.0036096d, E12=239°.961+0°.1643573d, E13=25°.053+12°.9590088d

7. Fig.2. Triangle used to define the direction of the angular velocity vector of geodetic rotation for any body of the Solar system.

8. Reduction formulae

9. The Earth σφσφ σθσθ σψσψ

10. The Earth Detail σφσφ σθσθ σψσψ

11. The Moon σφσφ σθσθ σψσψ

12. Mercury σφσφ σθσθ σψσψ

13. Pluto σφσφ σθσθ σψσψ

14. Table 3. The main secular and periodic terms of the geodetic rotation in longitude of the planetary equator ObjectSecular partPeriodic partEccentri- city of the orbit Mercury0.206 Venus 0.007 Earth0.017 Moon Mars0.093 Jupiter0.048 Saturn0.056 Uranus0.046 Neptune0.009 Pluto0.249 Sun

15. Venus σφσφ σθσθ σψσψ

16. Mars σφσφ σθσθ σψσψ

17. Jupiter σφσφ σθσθ σψσψ

18. Saturn σφσφ σθσθ σψσψ

19. Uranus σφσφ σθσθ σψσψ

20. Neptune σφσφ σθσθ σψσψ

21. The Sun σφσφ σθσθ σψσψ

22. Table 4. The secular terms of geodetic rotation for σ ψ, σ θ and σ φ Mercury213".3 +0".050T –0".029T 2 +… –0".036 +0".006T +6"·10 –4 T 2 +… 214".9 +0". 023T –0".029T 2 +… Jupiter0".312 –1"·10 –4 T +0".005T 2 +… 0".006 –3"·10 –4 T +5"·10 –5 T 2 +… 0".311 –1"·10 –4 T +0".005T 2 +… Venus43".048 –8"·10 –4 T+5"·10 –4 T 2 +… 0".741 –0".12T –0".002T 2 +… 43".091 –1"·10 –4 T+4"·10 –4 T 2 +… Saturn0".069 –4"·10 –4 T –0".001T 2 +… 0".003 +5"·10 –5 T –1"·10 –5 T 2 +… 0".06 –3"·10 –4 T –0".001T 2 +… The Earth 19".199 –4"·10 –4 T –0".001T 2 +… 1"·10 –5 +0".004T +0".012T 2 +… 17".615 –0.018T –9"·10 –4 T 2 +… Uranus0".012 +2"·10 –4 T +3"·10 –4 T 2 +… 2"·10 –4 –1"·10 –6 T +4"·10 –6 T 2 +… 0".002 +2"·10 –5 T +5"·10 –5 T 2 +… The Moon 19 ".494 +1"·10 –5 T –0 ".002T 2 +… 3"·10 –4 +0".004T +0".009T 2 +… 19 ".494 +9"·10 –6 T –0 ".002T 2 +… Neptune0".004 –3"·10 –5 T –4"·10 –5 T 2 +… 1"·10 –4 –1"·10 –6 T –1"·10 –6 T 2 +… 0".003 –2"·10 –5 T –4"·10 –5 T 2 +… Mars6".752 +2"·10 –5 T –0".026T 2 +… –0".12 +0".002T +6"·10 –4 T 2 +… 6".113 –0".007T –0".023T 2 +… Pluto0".002 +5"·10 –5 T +… 5"·10 –4 +1"·10 –5 T +… 0".001 +3"·10 –5 T +… The Sun7"·10 –4 –3"·10 –7 T +6"·10 –6 T 2 +… 2"·10 –6 –2"·10 –7 T +9"·10 –10 T 2 +… 7"·10 –4 –3"·10 –7 T +6"·10 –6 T 2 +… σψσθσφσψσθσφ σψσθσφσψσθσφ σψσθσφσψσθσφ σψσθσφσψσθσφ σψσθσφσψσθσφ σψσθσφσψσθσφ

23. Table 5. The periodic terms of geodetic rotation for σ ψ, σ θ and σ φ Mercury((1078".175 – 133".366T+...) cosλ 1 +(4845".763+35".597 T+...) sinλ 1 +...) ·10 –6 ((–0".182 +0".053T+...) cosλ 1 +(– 0".819 +0".128 T+...) sinλ 1 +...) ·10 –6 ((1086".284 –134".507T+...) cosλ 1 +(4882".206 +35".241T+...) sinλ 1 +...) ·10 –6 Venus((–56".807 +3".951T+...) cosλ 2 +(64".066 – 4".565T+...) sinλ 2 +...) ·10 –6 ((–0".978 +0".227T+...) cosλ 2 +(1".103 – 0".258T+...) sinλ 2 +...) ·10 –6 ((– 56".864 +3".954T+...) cosλ 2 +(64".13 – 4".569T+...) sinλ 2 +...) ·10 –6 The Earth ((–34".283 –7".54 T+...) cosλ 3 +(149".221 –5".682T+...) sinλ 3 +...) ·10 –6 ((3"·10 –5 –0".007T+...) cosλ 3 +(2"·10 –5 +0".03 T+...) sinλ 3 +...) ·10 –6 ((–31".454 –6".886T+...) cosλ 3 +(136".907 –5".348T+...) sinλ 3 +...) ·10 –6 The Moon ((–34".278 –7".559T+...) cosλ 3 +(149".2 –5".684T+...) sinλ 3 + (30".212 –0".001T+...) cos D +(–7"·10 –4 –0".001T +...) sin D +...) ·10 –6 ((–9"·10 –4 –0".008T+...) cosλ 3 +(3"·10 –4 +0".025 T+...) sinλ 3 + (–0".004 +0".01T+...) cos D +(–0".005 –0".007T +...) sin D +...) ·10 –6 ((–34".278 –7".559T+...) cosλ 3 +(149".2 –5".684T+...) sinλ 3 + (30".212 –0".001T +...) cos D +(–7"·10 –4 –0".001T +...) sin D +...) ·10 –6 Mars((515".779 +22".816T+...) cosλ 4 +(–229".128 +37".702T+...) sinλ 4 +...) ·10 –6 ((– 9".157 –0".241T+...) cosλ 4 +(4".068 –0".742T+...) sinλ 4 +...) ·10 –6 ((466".966 +20".149T+...) cosλ 4 +(–207".443 +34".359T+...) sinλ 4 +...) ·10 –6 σψσθσφσψσθσφ σψσθσφσψσθσφ σψσθσφσψσθσφ σψσθσφσψσθσφ σψσθσφσψσθσφ

24. Table 5. The periodic terms of geodetic rotation for σ ψ, σ θ and σ φ (c o n t i n u e d) Jupiter((82".801 +1".961T+...) cosλ 5 +(21".288+3".999T+...) sinλ 5 +...) ·10 –6 (( 1".586 –0".04 T+...) cosλ 5 +( 0".408 +0".057T+...) sinλ 5 +...) ·10 –6 ((82".699 +1".958T+...) cosλ 5 +(21".261+3".994T+...) sinλ 5 +...) ·10 –6 Saturn((–2".691 –5".173T+...) cosλ 6 +(52".94 –3".494T+...) sinλ 6 +...) ·10 –6 ((–0".113 –0".222T+...) cosλ 6 +( 2".24 –0".099T+...) sinλ 6 +...) ·10 –6 ((–2".362 –4".544T+...) cosλ 6 +(46".473 –3".022T+...) sinλ 6 +...) ·10 –6 Uranus((–22".266 –1".853T+...) cosλ 7 +(3".466 –0".881T+...) sinλ 7 +...) ·10 –6 (( –0".3 –0".018T+...) cosλ 7 +(0".047 –0".013T+...) sinλ 7 +...) ·10 –6 ((–3".011 –0".247T+...) cosλ 7 +(0".469 –0".12 T+...) sinλ 7 +...) ·10 –6 Neptune((1".839 +0".219T+...) cosλ 8 +(1".78 +0".302T+...) sinλ 8 +...) ·10 –6 ((0".057 +0".007T+...) cosλ 8 +(0".055 +0".009T+...) sinλ 8 +...) ·10 –6 ((1".627 +0".194T+...) cosλ 8 +(1".574 +0".267T+...) sinλ 8 +...) ·10 –6 Pluto((59".423 –2".028T+...) cosλ 9 +(–0".273–13".109 T+...) sinλ 9 +...) ·10 –6 ((16".174 –0".564T+...) cosλ 9 +(–0".076 – 3".564 T+...) sinλ 9 +...) ·10 –6 ((33".444 –1".119T+...) cosλ 9 +(–0".153 – 7".378 T+...) sinλ 9 +...) ·10 –6 σψσθσφσψσθσφ σψσθσφσψσθσφ σψσθσφσψσθσφ σψσθσφσψσθσφ σψσθσφσψσθσφ

25. Table 5. The periodic terms of geodetic rotation for σ ψ, σ θ and σ φ (c o n t i n u e d) The Sun((0".105 +0".003T+...) cosλ 5 +(0".027 +0".005T+...) sinλ 5 + (2"·10 –4 –3"·10 –5 T+...) cosλ 1 +(0".001 +8"·10 –6 T+...) sinλ 1 +...) ·10 –6 ((0".001 +8"·10 –5 T+...) cosλ 5 +(3"·10 –4 +6"·10 –5 T+...) sinλ 5 + (–1"·10 –5 +1"·10 –6 T+...) cosλ 1 +(–6"·10 –5 –2"·10 –6 T+...) sinλ 1 +...) ·10 –6 ((0".105 +0".003T+...) cosλ 5 +(0".027 +0".005T+...) sinλ 5 + (2"·10 –4 –3"·10 –5 T+...) cosλ 1 +(0".001 +8"·10 –6 T+...) sinλ 1 +...) ·10 –6 σψσθσφσψσθσφ λ 9 = 0.2480488137 + 25.2270056856T λ j (j=1,...9) – mean longitudes of the planets; λ 10 – mean geocentric longitude of the Moon; T – means the Dynamical Barycentric Time (TDB) measured in thousand Julian years (tjy) (of 365250 days) from J2000.

26. CONCLUSION For the Sun, giant planets and Pluto the geodetic rotation is insignificant. For the terrestrial planets and the Moon the geodetic rotation is significant and has to be taken into account for the construction of the high-precision theories of the rotational motion of these bodies. Geodetic rotation has to be taken into account if the influence of the dynamical figure of a body on its orbital-rotational motion is studied in the post-Newtonian approximation. The lunar laser ranging data processing has to use the relativistic theory of the rotation of the Moon, as well as that of the Earth.

27. R E F E R E N C E S 1.V.A..Brumberg, P.Bretagnon Kinematical Relativistic Corrections for Earth’s Rotation Parameters // in Proc. of IAU Colloquium 180, eds. K.Johnston, D. McCarthy, B. Luzum and G. Kaplan, U.S. Naval Observatory, 2000, pp. 293–302. 2.Seidelmann P.K., Archinal B.A., A'Hearn M.F., Cruikshank D.P., Hil-ton J.L., Keller H.U., Oberst J., Simon J.L., Stooke P., Tholen D.J., and Thomas P.C. (2005): Report of the IAU/IAG Working Group on Carto-graphic Coordinates and Rotational Elements: 2003, Celestial Mechanics and Dynamical Astronomy, 91, pp. 203-215. 3.Eroshkin G.I., Pashkevich V.V. (2007): Geodetic rotation of the Solar system bodies, Artificial Satellites, Vol. 42, No. 1, pp. 59–70.

28. A C K N O W L E D G M E N T S The investigation was carried out at the Central (Pulkovo) Astronomical Observatory of the Russian Academy of Science and the Space Research Centre of the Polish Academy of Science, under a financial support of the Cooperation between the Polish and Russian Academies of Sciences, Theme No 38.