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Astron. Astrophys. 329, 329-338 (1998)

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5. Form of the analytical solution and comparison with other solutions

The analytical solutions of the precession-nutation and rotation of the Earth are, for each variable, in the form of polynomials in time, Fourier series and Poisson series. The arguments of the Fourier and Poisson series are linear combinations of the angles arising in the analytical theories of the motion of the Moon, of the Sun, and of the planets used in this construction.

5.1. Theory of the Moon

The main part of the perturbations of the motion of the Moon is due to the Sun. These perturbations are represented with the 4 Delaunay angles D, F, l, [FORMULA] and we have the relations

[EQUATION]

in which [FORMULA], [FORMULA], [FORMULA] represent the mean longitudes of the Moon, of the perigee of the Moon, of the node of the Moon, respectively and [FORMULA] the mean longitude of the Earth.

The planetary perturbations in the solution of the motion of the Moon are represented with the 8 mean planetary longitudes [FORMULA] ([FORMULA]). Lastly, the perturbations of the motion of the Moon by the terrestrial equatorial bulge depend on the angle of precession [FORMULA] the frequency of which is [FORMULA] per year.

Therefore, we have on the whole 13 angles:

[EQUATION]

From linear combinations of such angles we find the arguments

[EQUATION]

of which the frequencies are [FORMULA] and [FORMULA] per year and the periods [FORMULA] and [FORMULA] years. There is no meaning to keep neither such arguments nor the angle [FORMULA], in the nutation theory which have to reach a high accuracy for a few ten years. Therefore, in the lunar theory, we have developed as polynomials in time and as Poisson series the arguments [FORMULA], [FORMULA] and [FORMULA].

Thus, the solution of the motion of the Moon is expressed as polynomials in time and as Fourier and Poisson series the arguments of which are linear combinations of 11 angles:

[EQUATION]

5.2. Theories of the Sun and the planets

The geocentric solutions of the motion of the Sun and the planets are also expressed as polynomials in time and as Fourier and Poisson series of the 11 angles (11), the angles D, F, l being due to the perturbations of the Earth-Moon barycenter (EMB) and of the planets by the Moon and also to the introduction of the motion of the Moon in the calculation of the vector EMB-Earth.

5.3. Theory of the precession-nutation and rotation of the Earth

The theory of the precession-nutation and rotation of the Earth built with the theories of the motion of the Moon, the Sun, and the planets described in the paragraphs above is also expressed with the 11 angles (11) and the angle of rotation of the Earth [FORMULA]

[EQUATION]

in which [FORMULA] and [FORMULA] are the values given in (5).

With this representation we do not find in the series neither arguments of very long period nor arguments with similar periods as it can be found in the classical nutation tables. For instance, we find in Souchay and Kinoshita (1997) the long period argument

[EQUATION]

and similar period arguments as

[FORMULA]

In each group, these terms are in phase every [FORMULA] years.

In the lunisolar nutation in longitude there also are similar period arguments

[FORMULA]

In pairs, these terms are in phase every [FORMULA] years or [FORMULA] years.

There is no meaning for keeping terms with periods of [FORMULA] years or several similar period terms in phase every [FORMULA] years or every [FORMULA] years or every [FORMULA] years because in the analytical solutions of the motion of the Moon, the Sun and the planets, such long period terms are developed as polynomials in time.

Moreover, there is no meaning for representing the solutions with arguments that it is impossible to discriminate over a 20 year time span of the high precision observations.

5.4. Comparison between SMART97 and other solutions

Roosbeek and Dehant (1997) have carried out comparisons between their solution, the solutions of Souchay and Kinoshita (1997), of Hartmann and Soffel (1994) and our solution. These comparisons display differences included between 500 and 1000 [FORMULA] as or more. This comes from errors that we have identified.

For instance, all the solutions except SMART97 have until now determined a contribution out-of-phase of the 18.6 year term with an amplitude of 135 [FORMULA] as. Their result is wrong by 250 [FORMULA] as (Bretagnon et al, 1997). In Kinoshita and Souchay (1990), this contribution out-of-phase was missing and produced an error of 384 [FORMULA] as (Bretagnon, 1996).

Another error common to the most of the other solutions is found in the calculation of the diurnal terms (24 hour, 12 hour, 8 hour period terms) of the precession-nutation of the rigid Earth:

a) the semidiurnal terms are out of phase by 30 degrees and therefore are wrong by 50%,

b) the 24 hour period terms, more important than the semidiurnal terms, are missing,

c) the 8 hour period terms are also missing.

We give in Table 4 a comparison between the Souchay-Kinoshita (1997) solution and SMART97 for the diurnal terms with an amplitude greater than 1 [FORMULA] as in the nutation in longitude. The 8 hour period terms do not appear in this table, the most important one having an amplitude of 0.14 [FORMULA] as.

[TABLE]

Table 4. Diurnal terms of the nutation in longitude. Comparison of the SMART97 solution with the Souchay-Kinoshita solution (SK97). Unit is [FORMULA] as (microarcsecond).

We have prefered to express the nutations as functions of the same angles (11) that the ones used in the analytical theories of the motion of the Moon, of the Sun and of the planets and we have not performed the transformation

[EQUATION]

where [FORMULA] is the longitude of the node of the Moon. For comparison, we give both forms of the arguments in Table 4.

We verify the absence of the out-of-phase parts in the Souchay-Kinoshita solution (SK97) due to the longitude ([FORMULA] West) of the major axis of the terrestrial equatorial ellipse. Moreover, it seems that, in SK97, two terms are wrong: the argument noted [FORMULA] is probably the argument [FORMULA] and the amplitude of the argument [FORMULA] is 0.31 [FORMULA] as instead of 5.0 [FORMULA] as. For the others semidiurnal terms, the differences of amplitude are less than 2.1 [FORMULA] as. Recently, the referee, H. Kinoshita, informed us that the last results of Souchay and Kinoshita, not yet published, seem to show a better agreement between SK97 and our solution, at one micro arcsecond level.

Lastly, let us note that it is useless to complete the precession-nutation solutions with the semidiurnal terms if one does not take into account the 24 hour period terms which are more numerous and the amplitude of which is more important.

5.5. Discussion about the choice of the angles

It is indifferent to perform or not to perform the transformation of the arguments (12) nevertheless it seems to us more advisable to keep the only angles introduced by the theories of the motion of the Moon, the Sun and the planets.

On the contrary, it is not correct to keep the longitude [FORMULA] of the perihelion of the Earth [FORMULA] in a periodic form. In this way the perihelion of the Earth reckoned from the equinox of date seems to have a period of [FORMULA] years when it includes many terms the most important of which have the following periods : [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] years. In the same way, it is not correct to keep the precession angle [FORMULA] in the nutation series. These terms with a period greater than [FORMULA] have to be developed as polynomials in time.

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© European Southern Observatory (ESO) 1998

Online publication: November 24, 1997
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