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

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

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

The Earth

σφ

σθ

σψ

The Earth Detail

σφ

σθ

σψ

The Moon

σφ

σθ

σψ

Mercury

σφ

σθ

σψ

Pluto

σφ

σθ

σψ

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

Venus

σφ

σθ

σψ

Mars

σφ

σθ

σψ

Jupiter

σφ

σθ

σψ

Saturn

σφ

σθ

σψ

Uranus

σφ

σθ

σψ

Neptune

σφ

σθ

σψ

The Sun

σφ

σθ

σψ

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.