determination of the geodetic rotation of the solar system bodies by means of spectral analysis...

Determination of the Geodetic Rotation of the Solar System Bodies by means of Spectral

Analysis MethodG.I. Eroshkin and V.V. Pashkevich

Central (Pulkovo) Astronomical Observatoryof Russian Academy of Science

St.PetersburgSpace Research Centre of Polish Academy of Sciences

Warszawa

2006

Here the index A means any point of Solar system or any Major body of Solar system; j – index for the summing all Major bodies of Solar system of the mathematical model of DE404/LE404 ephemeris; – gravitational constant;

– mass of the j -th body; c – velocity of light in vacuum; – distance between points A and j ; , , and –

barycentric vectors of the coordinate and velocity of these points; sign × is a vector product . If the point A is not a centre of the mass of the Sun the vector is practically orthogonal to the plane of the heliocentric orbit, so the mass of the Sun is the dominant mass of Solar system.

GjmjA jR

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�������������� AR

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The angular velocity vector of the geodetic rotation for any point of Solar system:

2 3

32 (1)

2j

A j A jAj A jA

GmR R R R

c

��������������������������������������������������������

A

The problem of the geodetic (relativistic) rotation of the Sun, major planets, and the Moon is studied by using DE404/LE404 ephemeris and by means of Spectral Analysis Method . For every of these bodies the files of the ecliptic components of the vector of the geodetic rotation were formed over time span from AD1000 to AD3000 at intervals of one day. Using the least-squares method and spectral analysis methods the secular and periodic components of the geodetic rotation vector were determined. The mean longitudes of the planets and the Moon adjusted to DE404/LE404 ephemeris were taken from Brumberg and Bretagnon (2000).

63 3 3

23 3 3 3

3

19".1988821 0".00035370T ... 10 34".285cos 149".227sin

T 7".539cos 5".682sin T 0".261cos 0".291sin ... ...

6283.0758511455 is the mean longitude of the Earth,

T means the time

z

(TDB) measured in thousand Julian years.

The Earth

For the Earth the component, orthogonal to the plane of the fixed ecliptic J2000.0 was determined:

This result is a very good agreement with that found analytically(Brumberg and Bretagnon, 2000). The method was applied to the other bodies of the solar system.

63 3 3

23 3 3 3

3

19".1988821 0".00035370T ... 10 34".285cos 149".227sin

T 7".539cos 5".682sin T 0".261cos 0".291sin ... ...

6283.0758511455 is the mean longitude of the Earth,

T means the time

z

(TDB) measured in thousand Julian years.

The Earth

For the Earth the component, orthogonal to the plane of the fixed ecliptic J2000.0 was determined:

This result is a very good agreement with that found analytically(Brumberg and Bretagnon, 2000). The method was applied to the other bodies of the solar system.

The Earth (in detail)

63 3 3

23 3 3 3

3

19".1988821 0".00035370T ... 10 34".285cos 149".227sin

T 7".539cos 5".682sin T 0".261cos 0".291sin ... ...

6283.0758511455

z

6

63 3 3 3 3

23 3

3

19".4950402 0".0000124T ... 10 30".212cos 0".001sin ... ...

10 34".280cos 149".204sin T 7".559cos 5".683sin

T 0".261cos 0".292sin ... ...

6283.0758511455 is the mean lon

z D D D

3

gitude of the Earth,

77713.7714481804, 180 , is the mean geocentric longitude of the Moon. D D

The Moon

6

63 3 3 3 3

23 3

3

19".4950402 0".0000124T ... 10 30".212cos 0".001sin ... ...

10 34".280cos 149".204sin T 7".559cos 5".683sin

T 0".261cos 0".292sin ... ...

6283.0758511455 is the mean lon

z D D D

3

gitude of the Earth,

77713.7714481804, 180 , is the mean geocentric longitude of the Moon. D D

The Moon

The Moon (in detail)

6

63 3 3 3 3

23 3

19".4950402 0".0000124T ... 10 30".212cos 0".001sin ... ...

10 34".280cos 149".204sin T 7".559cos 5".683sin

T 0".261cos 0".292sin ... ...

z D D D

3 6283.0758511455, 77713.7714481804D

Mercury

61 1 1 1 1

21 1

1

214".905 0".024T ...

10 1086".273cos 4882".196sin T 134".507cos 35".242sin

T 0".439cos 1".687sin ... ...

26087.903145742 is the mean longitude of Mercury.

z

Mercury

61 1 1 1 1

21 1

1

214".905 0".024T ...

10 1086".273cos 4882".196sin T 134".507cos 35".242sin

T 0".439cos 1".687sin ... ...

26087.903145742 is the mean longitude of Mercury.

z

Mercury (in detail)

61 1 1 1 1

21 1 1

214".905 0".024T ...

10 1086".273cos 4882".196sin T 134".507cos 35".242sin

T 0".439cos 1".687sin ... ... , 26087.903145742

z

Venus

62 2 2 2 2

22 2

2

43".12350 0".0011424T ...

10 56".907cos 64".182sin T 3".958cos 4".574sin

T 0".062cos 0".242sin ... ...

10213.2855462110 is the mean longitude of Venus.

z

Venus

62 2 2 2 2

22 2

2

43".12350 0".0011424T ...

10 56".907cos 64".182sin T 3".958cos 4".574sin

T 0".062cos 0".242sin ... ...

10213.2855462110 is the mean longitude of Venus.

z

Venus (in detail)

62 2 2 2 2

22 2

2

43".12350 0".0011424T ...

10 56".907cos 64".182sin T 3".958cos 4".574sin

T 0".062cos 0".242sin ... ...

10213.2855462110

z

Mars

64 4 4 4 4

24 4

4

6".755879 0".0002872T ...

10 516".062cos 229".326sin T 22".803cos 37".729sin

T 0".784cos 1".349sin ... ...

3340.6124266998 is the mean longitude of Mars.

z

Mars

64 4 4 4 4

24 4

4

6".755879 0".0002872T ...

10 516".062cos 229".326sin T 22".803cos 37".729sin

T 0".784cos 1".349sin ... ...

3340.6124266998 is the mean longitude of Mars.

z

Mars (in detail)

64 4 4 4 4

24 4 4

6".755879 0".0002872T ...

10 516".062cos 229".326sin T 22".803cos 37".729sin

T 0".784cos 1".349sin ... ... , 3340.6124266998

z

Jupiter

Saturn

Jupiter

Saturn

Jupiter (in detail)

Jupiter

Saturn

Saturn (in detail)

Uranus

Neptune

Pluto

Mercury (in detail)

61 1 1 1 1

21 1 1

214".905 0".024T ...

10 1086".273cos 4882".196sin T 134".507cos 35".242sin

T 0".439cos 1".687sin ... ... , 26087.903145742

z

2 32 (2)

| |j

j j

jj

GmR R

c R

���������������������������� ��������������

DISCUSSION

DE404/LE404 ephemeris is used for the definition of the geodetic rotation of the reference frame of DE404/LE404 ephemeris

– the angular velocity vector of the reference frame of DE404/LE404 ephemeris.

Here the index A means any point of Solar system or any Major body of Solar system; j – index for the summing all Major bodies of Solar system of the mathematical model of DE404/LE404 ephemeris; – gravitational constant;

– mass of the j -th body; c – velocity of light in vacuum; – distance between points A and j ; , , and –

barycentric vectors of the coordinate and velocity of these points; sign × is a vector product . If the point A is not a centre of the mass of the Sun the vector is practically orthogonal to the plane of the heliocentric orbit, so the mass of the Sun is the dominant mass of Solar system.

GjmjA jR

��������������AR

��������������jR

�������������� AR

��������������

The angular velocity vector of the geodetic rotation for any point of Solar system:

2 3

32 (1)

2j

A j A jAj A jA

GmR R R R

c

��������������������������������������������������������

A

Ecliptic components of the angular velocity vector of the Barycentre of Solar system

Ecliptic components of the angular velocity vector of the Barycentre of Solar system (in detail)

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. Landau L.D. and Lifshitz E.M., The Classical Theory of Fields: 1967, Moscow: "Nauka" , pp. 426-429. (in Russian)

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 Russian Academy of Science and the Space Research Centre of Polish Academy of Science, under a financial support of the Cooperation between Polish and Russian Academies of Sciences, Theme No 31.