7 The tidal acceleration of the Moon

The tidal component of the secular acceleration of the Moon's longitude is a fundamental parameter of the evolution of the Earth-Moon system. The tidal dissipation is due to a misalignment of the tidal bulge of the Earth relative to the Earth-Moon direction. This bulge exerts a secular torque and most of the effect comes from the ocean tides. It produces a secular negative acceleration of the Moon and a decrease in the Earth's rotation rate (increase of the length of day). A consequence of the negative acceleration in the lunar longitude of approximately $-25.8''/{\rm cy}^{2}$ is the well-known displacement of the Moon's barycenter that corresponds to an increase of the Earth-Moon distance of 3.8 cm/year.

Let us examine the quadratic terms of the lunar mean longitude W₁⁽²⁾ as it appears in the analytical lunar theory ELP2000-96 (Chapront et al. 1997). Here the constants and parameters are those from DE245, in particular Love numbers and time delays (see Sect. 8). W₁⁽²⁾ contains mainly 3 contributions (in $''/{\rm cy}^{2}$):
[1]       5.8665    Planetary perturbations;
[2]       0.1925    Earth's figure perturbations;
[3]  -12.8125    Tides.
These contributions have to be multiplied by 2 if we speak of "secular acceleration'' components.

In [1] the main effect is due to the secular variation of the solar eccentricity. It represents in fact a Taylor expansion of long periodic perturbations, mentioned also as "secular perturbations'' in classical celestial mechanics, with periods of the solar perigee and node of several ten thousand years.

In [2] the contribution is of the same nature. Hence, [1] and [2] are due to very long period effects. They have to be distinguished from the tides that induce dissipative forces. An analytical solution for the lunar motion also provides secular terms of higher degree in time (t³-terms, t⁴-terms, etc.). With the knowledge of all the secular components in the longitude, we are able to isolate the tidal acceleration from other perturbations.

We gather in Table 6 a non-exhaustive list of determinations of the tidal secular acceleration of the lunar longitude. The most recent values have been obtained with LLR observations. We note, for this type of determination, a significant improvement of the precision with more observations. It is also worth noticing that the most recent determination around $-25.86''/{\rm cy}^{2}$ comes closer to the value of Morrison & Ward (1975), $-26''/{\rm cy}^{2}$, obtained with an analysis of optical observations including occultations and planetary transits covering a time interval of approximately 2 centuries.

   

Table 6: Tidal acceleration of the lunar mean longitude (in arcsecond/cy²).

Authors	Value	Publication
Spencer Jonesa	-22	1939
Oesterwinter & Cohena	-38	1975
Morrison & Warda	-26	1975
Mullerb	-30	1976
Calame & Mulhollandc	-24.6	1978
Ferrari et al.d	-23.8	1980
Dickey et al.c	-23.8	1982
Dickey and Williamsc	-25.10	1982
Newhall et al.c	-24.90	1988
Chapront Touzé et al.c	-25.62	1997
(Solution S1998)c	-25.78	1999b
(Solution S2000)c	-25.836	2000
(this paper, solution 2001)c	-25.858

Type of observations: ^a Occultations, ^b eclipses, ^c LLR, ^d LLR and Lunar orbiter.

It is interesting to examine the intrinsic values of the tidal acceleration in various JPL numerical integrations (see Table 7). These values do not appear explicitly in the lists of parameters that are provided with each lunar ephemeris, but we have computed them from those parameters and from the models described in Sect. 8. It is worth noticing that the difference between the tidal secular acceleration in DE405 and our determination in S2001 is about $0.03''/{\rm cy}^{2}$ which gives an idea of the present uncertainty for this fundamental lunar parameter.

 

 

Table 7: Tidal acceleration of the lunar mean longitude in various JPL ephemerides (in arcsecond/cy²) (Standish 1982, 1995, 1998).

JPL ephemeris	Value	Publication
DE200	-23.895	1982
DE245	-25.625	1990
DE403	-25.580	1995
DE405	-25.826	1998

The expression of the lunar mean longitude of the Moon W₁ has the following secular expansion: W₁ = W₁⁽⁰⁾ + W₁⁽¹⁾ t + W₁⁽²⁾ t² +..., where t is the time in century reckoned from J2000.0; W₁⁽⁰⁾ is the constant term; W₁⁽¹⁾ is the sidereal mean motion for J2000.0; W₁⁽²⁾ is the total half secular acceleration of the Moon. We have investigated the convergence of these quantities to the S2001 values given in Table 4, when the upper limit of the time interval of observations increases. The results are given in Fig. 3 where $\Delta W_1^{(0)}$, $\Delta W_1^{(1)}$ and $\Delta W_{1}^{(2)}$ are the corrections to S2001 values obtained in the intermediate solutions (1996-2001). In particular the variation of $\Delta W_{1}^{(2)}$ shows the evolution of the fitted value of the tidal acceleration when using more and more recent LLR observations: as mentioned above W₁⁽²⁾ is the sum of several contributions listed [1], [2] and [3] at the beginning of the present section, and we may assume that [1] and [2] have been computed with sufficient accuracy through the secular terms in ELP theory; only [3] (half tidal acceleration) needs to be fit. This graph allows us to ensure nowadays a realistic precision in the knowledge of the secular acceleration of better than $0.03''/{\rm cy}^{2}$, in agreement with the conclusion resulting from the comparison of the S2001 value to the DE405 one.

Figure 3: Time evolution of the corrections $\Delta$ to the secular components of the mean longitude of the Moon, W1 = W1(0) + W1(1) t + W1(2) t2, when increasing the set of LLR measurements.