A&A 400, 785-790 (2003)
DOI: 10.1051/0004-6361:20021912
P. Bretagnon^{} - A. Fienga - J.-L. Simon
Institut de Mécanique Céleste, Observatoire de Paris, UMR 8028 du CNRS, 77 avenue Denfert-Rochereau, 75014 Paris, France
Received 3 June 2002 /Accepted 29 November 2002
Abstract
Since the adoption by the XXIVth General Assembly of an
accurate model of nutation,
the IAU
encourages the development of new expressions
for precession consistent with the new model. This paper presents new expressions
for the precession quantities issued from the analytical solution of the rotation
of the rigid Earth SMART97 (Bretagnon et al. 1998) which provides together
the precession and the nutation. These expressions include the new value
of the precession rate of the equator in longitude.
As the SMART97 series are close to the
Souchay et al. (1999) series used to build the new model, our expressions are consistent
with the IAU 2000 Precession-Nutation Model. In the other parts of the paper,
we discuss some concepts. In Sect. 3, we propose the definition
of a conventional ecliptic plane close to the mean ecliptic J2000 and with a
non-rotating origin. Section 4 deals with the
Earth Orientation Parameters. We show that the celestial pole offsets, as
well as the polar motion can be described with the Euler's angles.
At last, in Sect. 5, we give some recommendations about the
precession-nutation variables and the arguments of the series of
the nutation.
Key words: celestial mechanics - Earth
These nutation series do not provide expressions for precession and for this reason the IAU "encourages the development of new expressions for precession consistent with the IAU 2000A model'' (encouragement 3 of resolution B1.6). Meanwhile, the IERS Conventions 2000 (McCarthy 2002) recommend the use of the precessional formulae derived from Lieske et al. (1977) with improved numerical values for the precession rate of the equator in longitude and obliquity. These formulae are based upon the use of the secular variations of the ecliptic pole from Newcomb's theory of the Sun and of old value of the precession constant and of the masses of the planets. Up to date developments have been computed by Simon et al. (1994) and Williams (1994) but they are not really issued from a new theory of the precession-nutation.
In this paper we present, at first, new expressions for the precession quantities, issued from the analytical solution of the rigid Earth SMART97 (Bretagnon et al. 1998) which provides together developments for the precession and series for the nutation. These expressions are consistent with the IAU 2000A model and are more precise than the ones given by Lieske et al. In a second part, we propose and discuss the concept of a conventional ecliptic. In a third part, we introduce some ideas concerning the Earth Orientation Parameters (EOP) in connection with the Euler's angles. At last, we give some considerations on the precession-nutation variables and on the arguments of the series of the nutation.
Figure 1: Ecliptic and equator J2000 and of date. | |
Open with DEXTER |
(1) | |||
The value of
corresponds, for the precession rate
of the equator in longitude,
to the value
(2) |
(3) |
This paper | IAU | ||
Precession | Nutation | Precession | Nutation |
(4) |
(5) |
t^{0} | t | t^{2} | t^{3} | t^{4} | t^{5} | t^{6} | t^{7} | |
P_{A} | 0.00000 | 41.99604 | 19.39715 | - 0.22350 | - 0.01035 | 0.00019 | 0.0 | 0.0 |
Q_{A} | 0.00000 | -468.09550 | 5.10421 | 0.52228 | - 0.00569 | - 0.00014 | 0.00001 | 0.0 |
0.00000 | 469.97560 | - 3.35050 | - 0.12370 | 0.00030 | 0.0 | 0.0 | 0.0 | |
0.0 | 0.0 | |||||||
0.00000 | 50287.92262 | 111.24406 | 0.07699 | - 0.23479 | -0.00178 | 0.00018 | 0.00001 | |
0.00000 | 20041.90936 | -42.66980 | -41.82364 | - 0.07291 | - 0.01127 | 0.00036 | 0.00009 | |
2.72767 | 23060.80472 | 30.23262 | 18.01752 | - 0.05708 | - 0.03040 | -0.00013 | 0.0 | |
z_{A} | - 2.72767 | 23060.76070 | 109.56768 | 18.26676 | - 0.28276 | - 0.02486 | -0.00005 | 0.0 |
84381.40880 | - 468.36051 | - 0.01667 | 1.99911 | - 0.00523 | - 0.00248 | -0.00003 | 0.0 | |
84381.40880 | - 0.26501 | 5.12769 | - 7.72723 | - 0.00492 | 0.03329 | -0.00031 | -0.00006 | |
0.00000 | 50384.78750 | - 107.19530 | - 1.14366 | 1.32832 | - 0.00940 | -0.00350 | 0.00017 | |
0.00000 | 105.57686 | - 238.13769 | - 1.21258 | 1.70238 | - 0.00770 | -0.00399 | 0.00016 |
t^{0} | t | t^{2} | t^{3} | t^{4} | t^{5} | t^{6} | t^{7} | Difference | |
over 100 yrs | |||||||||
P_{A} | 0.00000 | 0.02004 | -0.04985 | - 0.0445 | - 0.01035 | 0.00019 | 0.0 | 0.0 | 0.003 |
Q_{A} | 0.00000 | 0.05450 | 0.04521 | 0.17828 | - 0.00569 | - 0.00014 | 0.00001 | 0.0 | 0.006 |
0.00000 | -0.05340 | - 0.04850 | - 0.18370 | 0.00030 | 0.0 | 0.0 | 0.0 | 0.006 | |
0.0 | 0.0 | ||||||||
0.00000 | -0.04688 | 0.13106 | 0.08299 | - 0.23479 | -0.00178 | 0.00018 | 0.00001 | 0.006 | |
0.00000 | -1.19964 | -0.00480 | 0.00936 | - 0.07291 | - 0.01127 | 0.00036 | 0.00009 | 0.120 | |
2.72767 | -1.37628 | 0.04462 | 0.01952 | - 0.05708 | - 0.03040 | -0.00013 | 0.0 | 0.138 | |
z_{A} | - 2.72767 | -1.42030 | 0.09968 | 0.06376 | - 0.28276 | - 0.02486 | -0.00005 | 0.0 | 0.143 |
-0.03920 | -0.21051 | 0.04233 | 0.18611 | - 0.00523 | - 0.00248 | -0.00003 | 0.0 | 0.021 | |
-0.03920 | - 0.01261 | 0.00069 | -0.00123 | - 0.00492 | 0.03329 | -0.00031 | -0.00006 | 0.001 | |
0.00000 | 0.00000 | 0.06370 | 0.00334 | 1.32832 | - 0.00940 | -0.00350 | 0.00017 | 0.001 | |
0.00000 | 0.05086 | -0.07369 | -0.08758 | 1.70238 | - 0.00770 | -0.00399 | 0.00016 | 0.006 |
The expression for Greenwich Mean Sidereal Time (GMST) at ,
coming from
is
= | |||
(6) | |||
The polynomial part of the third Euler's angle which measures the diurnal rotation of the Earth,
coming from the relation
,
is
= | 4.894 961 212 82+2 301 216.753 652 525 t | (7) | |
-0.000 476 710 t^{2}-0.000 005 892 t^{3}+0.000 006 821 t^{4} | |||
-0.000 000 044 t^{5}-0.000 000 018 t^{6} +0.000 000 001 t^{7} |
From the formulae of Simon et al., we can compute
the corrections
of the precession quantities X
given by future improvements
and
of
and
.
These corrections have the form
(8) |
= | 0 | ||
= | 0 | ||
= | 0 | ||
= | 0 | ||
= | |||
= | |||
= | |||
= | |||
= | (9) | ||
= | |||
= | |||
= |
1) a rotation about the z axis of
with
(10) |
The angles , and which entirely define the rotation of the Earth will be used for in the conventional ecliptic from the origin of the ICRF, for from the conventional ecliptic and for from the intersection of the true equator of date and the conventional ecliptic.
In the same manner as new expressions for precession were given in SMART (Sect. 2), a simplification of the implementations of the Earth Orientation Parameters (EOP) can be proposed for basic users. They describe the observed Earth variations of rotation, the observed offsets in the pole positions and polar motion by the means of the same Euler angles used in SMART. The plane of reference is a fixed plane, the conventional ecliptic, described in Sect. 3.
The EOP are the 5 Earth orientation parameters obtained by comparison of the current Earth rotation models and IERS observations. The parameters are the differences (UT1-UTC) or (UT1-TAI), the coordinates of the terrestrial pole and the celestial pole offsets (IERS Conventions, 1996).
The celestial pole offsets are observed variations of the celestial pole positions
induced by mismodeling in precession and nutation theories in the
International Celestial Reference Frame.
Till the new 2000 IERS Conventions, these observed variations were given as differences
in longitude and in obliquity referred to the equator and
the equinox of date (,
), (noted as (
,
)
by IERS)
with respect to the conventional celestial pole coordinates defined by models (IERS Conventions, 1996).
However, as it is recommended in the IERS Conventions (2000), such offsets between observed
coordinates of the celestial pole (
,
)
and computed ones must be
provided as (X, Y) corrections where:
= | |||
= |
After data analysis, IERS also provides the differences between UT1 and TAI or UT1 and UTC.
These differences describe the variations of the Earth rotation.
(UT1-UTC) published by IERS is connected to the sideral time ST (IERS Conventions, 1996) with:
The last EOP are the coordinates of the celestial pole relative to the International Terrestrial Reference Frame (ITRF). After correcting for the celestial pole offsets and the variations of the Earth rotation, a residual rotation remains. It corresponds to pole position variations caused by free Chandler wobble and induced by atmospheric and oceanic mass redistribution. These variations, that have amplitudes of several tenths of arcseconds and quasidiurnal period in Euler angles, could also be seen as differences between the coordinates of the celestial pole in a rotating frame at the time of observation and the observed coordinates of the terrestrial pole in the ITRF. Therefore, they are associated with two small angles (x, y) defined as the celestial pole coordinates relative to the International Terrestrial Reference Frame (ITRF).
To describe the EOP in Euler angles, we note (,
)
two angles of a small rotation
in the equatorial J2000 frame.
This rotation could also be written as small variations of Euler angles (, ).
= | |||
= | |||
Furthermore, in Bretagnon et al. (1997), the angle of the Earth rotation is also given
as an Euler angle ,
measured from the intersection of the equator of date and the ecliptic J2000.
is connected to the sideral time, ST, measured from the equinox of date, by the relation:
As it was described in the previous subsection, the EOP are published as small variations of the pole coordinates in the GCRF and ITRF and variations of angles of the Earth rotation. It is then possible to transform these small rotations in Euler angles in following the equations system 12 and 13. We will then obtain 3 Euler angles representing the EOP, and a global rotation could then be estimated.
If we note (
,
), the rotation induced by the pole offsets
(in a non-rotating frame), (
,
), the rotation from the polar motion
(in a rotating frame) and ,
the variation of the Earth rotation,
then the global rotation induced by the variations of the reference pole coordinates in GCRF and ITRF
and by the variations of the Earth rotation can be described by the Euler angles
(
,
,
)
such as:
= | |||
= | |||
= | r.(UT1-UTC). |
As it was demonstrated and discussed by several authors (Capitaine 1990; Bizouard 1995), the main difficulty of the representation of the EOP as Euler angles is the commensurability of the observed periods used to estimate the EOP.
The polar coordinates variations or polar motion ( , ) are detected from observations with periods smaller than 2 days, but the celestial pole offsets ( , ) are estimated after analysis of multi-day observations. If we include in the Euler angles the polar motion and the pole offsets, then we will mix effects which have different frequency domains. Such questions are quite complex, but new results (Richter 2001) tend to demonstrate that the correlation problems seem to be solved in using very dense sets of observations. Nevertheless, in waiting for a more definitive conclusion, it is still possible, after the computation of the 5 classical EOP by comparison to observations, to publish for the basic users the EOP as three Euler angles. Such publication, containing Euler angles given with tabulated coefficients, will make the EOP easier to use for astronomers who are not specialists in Earth rotation and who do not want to do geophysical studies, but who want to reduce very accurate astrometric observations in order to obtain very precise orbits or positions of solar system objects, stars, quasars, etc. However, as it was stressed by Rothacher (2001), such tabulated publication would face the problem of the high number of coefficients needed to allow linear or quadratic interpolations. Studies must be lead to estimate the frequency at which the Euler angles would have to be estimated and the amount of data needed for the publications. The use of the non-rigid SMART solution associated with the representation of the EOP as Euler angles will give a consistent model for the Earth rotation in the GCRF and ITRF.
The variables , and z defining the mean equator of date with respect to the equator J2000 cannot represent the precession-nutation because they are singular in the time origin. They are therefore useless.
It is thus necessary to describe nutation as Fourier and Poisson series whose angles are linear combinations of 12 arguments. These arguments are the 8 mean longitudes of planets reckoned from the origin of the ICRF, the angles of Delaunay D, F, l independent of the origin and the angle of rotation of the Earth about its axis.
In any case, no argument must be reckoned from the equinox of date (Bretagnon 1998).
For the adopted value of the precession constant
Taking into account the adoption of the ICRF by IAU in 1997, we propose to define a conventional ecliptic frame obtained from the ICRF by a rotation about the x axis of (Eq. (10)). This conventional ecliptic will be the reference frame in which the analytical theories of the motion of planets will be built. It is also with respect to this reference frame that the Euler's angles must be defined.
After the construction of a non-rigid SMART solution using the transfer function MHB2000 of Mathews et al., we propose to provide a precession-nutation solution expressed in the Euler variables and containing the diurnal and subdiurnal terms. We propose also an alternative representation of the Earth Orientation Parameters as corrections of the Euler's angles.
Acknowledgements
We wish to thank Dr. J. Vondrak for his useful comments and suggestions.