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On the geodetic rotation of the major planets, the Moon and the
Sun G.I. Eroshkin and V.V. Pashkevich
Central (Pulkovo) Astronomical Observatoryof the Russian Academy of Science
St.PetersburgSpace Research Centre of the Polish Academy of Sciences
Warszawa
2009
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:
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 .
jm
The angular velocity vector of the geodetic rotation for any body of Solar system:
32
1 32 .
2j
i i j i jj i
i j
GmR R R R
c R R
, , ,i i j jR R R R
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)
Mercury α0=281°.01-0°.033T δ0=61°.45-0°.005T W=329°.548+6°.1385025d
Saturn α0=40°.589-0°.036Tδ0=83°.537-0°.004T W=38°.90+810°.7939024d
Venus α0=272°.76 δ0=67°.16 W=160°.20-1°.4813688d
Uranus α0=257°.311 δ0=-15°.175 W= 203°.81-501°.1600928d
The
Earth α0=0°.00-0°.641T δ0=90°.00-0°.557T W=190°.147+360°.9856235d
Neptune α0=299°.36+0°.70 sin N δ0=43°.46-0°.51 cos N W=253°.18+536°.3128492d -0°.48 sin N N=357°.85+52°.316T
Mars α0=317°.68143-0°.1061T δ0=52°.88650-0°.0609T W=176°.630+350°.89198226d
Pluto α0= 313°.02δ0=9°.09 W=236°.77-56°.3623195d
Jupiter α0=268°.05-0°.009T δ0=64°.49+0°.003T W=284°.95+870°.5366420d
The Sun α0=286°.13 δ0=63°.87 W=84°.10+14°.1844000d d - interval in days from J2000;
T - interval in Julian centuries (of 36525 days) from J2000.
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
α0=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 E13δ0= 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 E13W=38°.3213 +13°.17635815d -1°.4 x 10-12d2 +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
Fig.2. Triangle used to define the direction of the angular velocity vector of geodetic rotation for any body of the Solar system.
Reduction formulae
0 0
0 0 0
0 0 0
0 0
0 0
cos sin
sin cos
Ecl EquX X
Ecl Equ EquY Y Z
Ecl Equ EquZ Y Z
* 0 0 0 0 0
0 0
*
cos cos sin sin sin cos
cos cossin
sin
* 0 0
* 0 0
* 0
cos sin
sin cos
Ecl Ecl EclX X Y
Ecl Ecl EclY X Y
Ecl EclZ Z
*
*
* ** *sin cos
EclZ
EclX
Ecl EclY Z
6 61 11078".175 10 sin 4845".763 10 cos
6 62 256".807 10 sin 64".066 10 cos
6 63 334".278 10 sin 149".200 10 cos
6 63 334".283 10 sin 149".221 10 cos
630".212 10 sin D 6 6
4 4515".779 10 sin 229".128 10 cos 6 6
5 582".801 10 sin 21".288 10 cos 6 6
6 62".691 10 sin 52".940 10 cos 6 6
7 722".266 10 sin 3".466 10 cos 6 6
8 81".839 10 sin 1".780 10 cos 6 6
9 959".423 10 sin 0".273 10 cos
610".001 10 cos
6 65 50".105 10 sin 0".027 10 cos
Table 3. The main secular and periodic terms of the geodetic rotation in
longitude of the planetary equator
213".300T
43".048T
19".199T
19".494T
6".752T
0".312T
0".069T
0".012T0".004T
0".002T
0".001T
Object Secular part Periodic part Eccentri-city of
the orbit
Mercury 0.206
Venus 0.007
Earth 0.017
Moon
Mars 0.093
Jupiter 0.048
Saturn 0.056
Uranus 0.046
Neptune 0.009
Pluto 0.249
Sun
Table 4. The secular terms of geodetic rotation for σψ, σθ and σφ
Mercury 213".3 +0".050T –0".029T2+…
–0".036 +0".006T +6"·10–4 T2+…
214".9 +0". 023T –0".029T2+…
Jupiter 0".312 –1"·10–4 T +0".005T2+…
0".006 –3"·10–4 T +5"·10–5T2+…
0".311 –1"·10–4 T +0".005T2+…
Venus 43".048 –8"·10–4 T+5"·10–4 T2+…
0".741 –0".12T –0".002T2+…
43".091 –1"·10–4 T+4"·10–4 T2+…
Saturn 0".069 –4"·10–4 T –0".001T2+…
0".003 +5"·10–5 T –1"·10–5T2+…
0".06 –3"·10–4 T –0".001T2+…
The Earth
19".199 –4"·10–4 T –0".001T2+…
1"·10–5 +0".004T +0".012T2+…
17".615 –0.018T –9"·10–4 T2+…
Uranus 0".012 +2"·10–4 T +3"·10–4T2+…
2"·10–4 –1"·10–6 T +4"·10–6 T2 +…
0".002 +2"·10–5 T +5"·10–5T2+…
The Moon
19 ".494 +1"·10–5 T –0 ".002T2+…
3"·10–4 +0".004T +0".009T2+…
19 ".494 +9"·10–6 T –0 ".002T2+…
Neptune 0".004 –3"·10–5 T –4"·10–5T2 +…
1"·10–4 –1"·10–6 T –1"·10–6T2 +…
0".003 –2"·10–5 T –4"·10–5T2 +…
Mars 6".752 +2"·10–5 T –0".026T2+…
–0".12 +0".002T +6"·10–4 T2+…
6".113 –0".007T –0".023T2+…
Pluto 0".002 +5"·10–5 T +…
5"·10–4 +1"·10–5 T +…
0".001 +3"·10–5 T +…
The Sun 7"·10–4 –3"·10–7 T +6"·10–6 T2+…
2"·10–6 –2"·10–7 T +9"·10–10 T2+…
7"·10–4 –3"·10–7 T +6"·10–6 T2+…
σψ
σθ
σφ
σψ
σθ
σφ
σψ
σθ
σφ
σψ
σθ
σφ
σψ
σθ
σφ
σψ
σθ
σφ
Table 5. The periodic terms of geodetic rotation for σψ, σθ and σφ
Mercury ((1078".175 – 133".366T+...) cosλ1 +(4845".763+35".597 T+...) sinλ1+...) ·10–6
((–0".182 +0".053T+...) cosλ1 +(– 0".819 +0".128 T+...) sinλ1 +...) ·10–6
((1086".284 –134".507T+...) cosλ1 +(4882".206 +35".241T+...) sinλ1 +...) ·10–6
Venus ((–56".807 +3".951T+...) cosλ2 +(64".066 – 4".565T+...) sinλ2 +...) ·10–6
((–0".978 +0".227T+...) cosλ2 +(1".103 – 0".258T+...) sinλ2 +...) ·10–6
((– 56".864 +3".954T+...) cosλ2 +(64".13 – 4".569T+...) sinλ2 +...) ·10–6
The Earth
((–34".283 –7".54 T+...) cosλ3 +(149".221 –5".682T+...) sinλ3 +...) ·10–6
((3"·10–5 –0".007T+...) cosλ3 +(2"·10–5 +0".03 T+...) sinλ3 +...) ·10–6
((–31".454 –6".886T+...) cosλ3 +(136".907 –5".348T+...) sinλ3 +...) ·10–6
The Moon
((–34".278 –7".559T+...) cosλ3 +(149".2 –5".684T+...) sinλ3 +
(30".212 –0".001T+...) cos D +(–7"·10–4 –0".001T +...) sin D +...) ·10–6
((–9"·10–4 –0".008T+...) cosλ3 +(3"·10–4 +0".025 T+...) sinλ3 +
(–0".004 +0".01T+...) cos D +(–0".005 –0".007T +...) sin D +...) ·10–6
((–34".278 –7".559T+...) cosλ3 +(149".2 –5".684T+...) sinλ3 +
(30".212 –0".001T +...) cos D +(–7"·10–4 –0".001T +...) sin D +...) ·10–6
Mars ((515".779 +22".816T+...) cosλ4 +(–229".128 +37".702T+...) sinλ4 +...) ·10–6
((– 9".157 –0".241T+...) cosλ4 +(4".068 –0".742T+...) sinλ4 +...) ·10–6
((466".966 +20".149T+...) cosλ4 +(–207".443 +34".359T+...) sinλ4 +...) ·10–6
σψ
σθ
σφ
σψ
σθ
σφ
σψ
σθ
σφ
σψ
σθ
σφ
σψ
σθ
σφ
Table 5. The periodic terms of geodetic rotation for σψ, σθ and σφ (c o n t i n u e d)
Jupiter ((82".801 +1".961T+...) cosλ5 +(21".288+3".999T+...) sinλ5 +...) ·10–6
(( 1".586 –0".04 T+...) cosλ5 +( 0".408 +0".057T+...) sinλ5 +...) ·10–6
((82".699 +1".958T+...) cosλ5 +(21".261+3".994T+...) sinλ5 +...) ·10–6
Saturn ((–2".691 –5".173T+...) cosλ6 +(52".94 –3".494T+...) sinλ6 +...) ·10–6
((–0".113 –0".222T+...) cosλ6 +( 2".24 –0".099T+...) sinλ6 +...) ·10–6
((–2".362 –4".544T+...) cosλ6 +(46".473 –3".022T+...) sinλ6 +...) ·10–6
Uranus ((–22".266 –1".853T+...) cosλ7 +(3".466 –0".881T+...) sinλ7 +...) ·10–6
(( –0".3 –0".018T+...) cosλ7 +(0".047 –0".013T+...) sinλ7 +...) ·10–6
((–3".011 –0".247T+...) cosλ7 +(0".469 –0".12 T+...) sinλ7 +...) ·10–6
Neptune ((1".839 +0".219T+...) cosλ8 +(1".78 +0".302T+...) sinλ8+...) ·10–6
((0".057 +0".007T+...) cosλ8 +(0".055 +0".009T+...) sinλ8 +...) ·10–6
((1".627 +0".194T+...) cosλ8 +(1".574 +0".267T+...) sinλ8 +...) ·10–6
Pluto ((59".423 –2".028T+...) cosλ9 +(–0".273–13".109 T+...) sinλ9 +...) ·10–6
((16".174 –0".564T+...) cosλ9 +(–0".076 – 3".564 T+...) sinλ9 +...) ·10–6
((33".444 –1".119T+...) cosλ9 +(–0".153 – 7".378 T+...) sinλ9 +...) ·10–6
σψ
σθ
σφ
σψ
σθ
σφ
σψ
σθ
σφ
σψ
σθ
σφ
σψ
σθ
σφ
Table 5. The periodic terms of geodetic rotation for σψ, σθ and σφ (c o n t i n u e d)
The Sun ((0".105 +0".003T+...) cosλ5 +(0".027 +0".005T+...) sinλ5 +
(2"·10–4 –3"·10–5 T+...) cosλ1 +(0".001 +8"·10–6 T+...) sinλ1 +...) ·10–6
((0".001 +8"·10–5 T+...) cosλ5 +(3"·10–4 +6"·10–5 T+...) sinλ5 +
(–1"·10–5 +1"·10–6 T+...) cosλ1 +(–6"·10–5 –2"·10–6 T+...) sinλ1 +...) ·10–6
((0".105 +0".003T+...) cosλ5 +(0".027 +0".005T+...) sinλ5 +
(2"·10–4 –3"·10–5 T+...) cosλ1 +(0".001 +8"·10–6 T+...) sinλ1 +...) ·10–6
σψ
σθ
σφ
1 4.40260867435 26087.9031415742 T.
2 3.17614652884 10213.2855462110T
3 1.75347029148 6283.0758511455T .
5.19846640063 77713.7714481804 T .D 4 6.20347594486 3340.6124266998T .
5 0.59954632934 529.6909650946T
6 0.87401658845 213.2990954380T
7 5.48129370354 74.7815985673T
8 5.31188611871 38.1330356378T
λ9 = 0.2480488137 + 25.2270056856T
10 3 180 ,D
λj (j=1,...9) – mean longitudes of the planets; λ10 – mean geocentric longitude of the Moon; T – means the Dynamical Barycentric Time (TDB) measured in thousand Julian years (tjy) (of 365250 days) from J2000.
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.
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.
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.