Determination of the free lunar libration modes from ephemeris DE430
1 Introduction
There are many moons which keep facing one side of their central planet on average in solar system bodies, including Earth’s Moon, most regular satellites of the giant planets and even Pluto’s moon Charon. This is called synchronous rotation, which was originally described in work by Cassini in the eighteenth century. For the Moon, because of the aspherical distribution of lunar mass, the rotational dynamics are not uniform. There are two types of librations: forced libration and free libration; the former is caused by timevarying torque on the lunar figure due to attraction by the Earth, Sun and planets, and the latter is a kind of normal mode excited by internal or external activities.
There are three modes in free librations for a solid Moon, and the periods can be calculated by linear theory (Eckhardt 1965 ; Eckhardt 1981 ). The first and second modes, the longitudinal and latitudinal modes respectively, are equivalent to the length of day variation and the nutation in the Earth, and have periods of about 2.9 and 81 yr, respectively. The third mode, on the other hand, is like the Chandler wobble in the Earth and is called the wobble mode with a 75 yr period. The free librations are different from the forced librations where amplitudes, phases and periods can all be calculated theoretically from a lunar structure model. Only the periods can be calculated for the free librations, and observations are necessary for determination of the amplitudes.
Due to energy dissipation, the free librations damp out over time. If the free librations exist, there must be some sources of excitation such as impacts that recently occurred. Analyses of Lunar Laser Ranging (LLR) data for detection of free librations have already been performed by some researchers (Calame 1977 ; Jin & Li 1996 ; Chapront et al. 1999 ; Newhall & Williams 1996 ; Rambaux & Williams 2011 ), and the amplitudes and phases have been determined.
The results of previous studies were reflected in the ephemerides DE403 (Standish 1995 ) and DE421 (Folkner et al. 2009 ). The present paper extracts free modes from DE430 (Folkner et al. 2014 ), which is reported to be the most accurate lunar ephemeris today. Compared with DE421, DE430 added new data for the Moon and planets, for example, estimations of the lunar orbit and rotation have been improved through usage of additional LLR data (about 5 yr), i.e. about more than 14% data that were used in the fit, and also an unprecedentedly improved gravity field model from the Gravity Recovery and Interior Laboratory (GRAIL) mission (Williams et al. 2014 ) was used to create the ephemeris. The free librations depend on the orbital and lunar physical model and also the method of data fit. With the improvement of models and fits with time, we expect estimations of the free librations embedded in the Euler angles to also improve. Therefore, as a preliminary study and first step toward a better understanding of lunar rotation, we try to extract the free libration modes from DE430.
In this paper, newly released lunar ephemeris DE430 is analyzed to extract the free libration modes, and we employ the Fourier analysis method (Newhall & Williams
1996
) with our improvement to determine the periods of the librations, and further use the leastsquares method to estimate amplitudes of the librations. The trace of pole vectors with direction cosines
2 Libration Model
Study of the rotation of the Moon is based on the following three laws provided by Cassini (Beletskii
1972
):
The Moon rotates about its polar axis with constant angular velocity equal to that of its revolution around Earth.
The inclination of the Moonʼs equator with respect to the plane of the ecliptic is constant.
The poles of the Moonʼs axis of rotation, those of the ecliptic, and those of the lunar orbit, lie in one great circle.
When we use the selenocentric coordinate system, it is supposed that the Moon has the form of a triaxial ellipsoid, with its longest axis directed towards the mean direction of the Earth, and the smallest one is the axis of rotation. We direct OX
_{1} along the longest axis, OX
_{3} along the axis of rotation with positive being northwards and OX
_{2} so as to form a righthanded coordinate system. The directions of these axes with respect to the mean Earth equatorial system at J2000.0
If Cassiniʼs laws hold perfectly when applied, the Euler angles would satisfy the equations
Usually, σ is multiplied by I to be comparable to ρ, ρ and
Time variations of the libration angles and polar coordinates are shown in Figures.
3 Free libration of the Moon
The lunar physical librations have three components of free modes: two components of the pole, P _{1} and P _{2}, and one longitudinal τ. In order to describe lunar rotation, the Moon is usually considered as a solid body, and the elastic deformation and presence of the core alter the proper periods by only small amounts (Williams et al. 2001 ). In this case, P _{1},P _{2} and τ which are functions of time can be determined from the well known EulerLiouville equations,
We get the linear approximation of the differential equations as Rambaux and Williams ( 2011 ), and add a small correction term (Moons 1982 ) to the third one,
The longitude frequency is obtained as


4 Frequency analysis
The forced and free lunar physical librations are included together in the ephemeris. Firstly, we obtain the Euler angles from DE430 covering 1100 yr (from 1550 A.D. to 2649 A.D.), then we transform the angles from the equatorial reference frame with equinox at J2000.0 to the ecliptic of date. We use I=0.02692 radian and Ω, λ from Simon et al.
1994
. To estimate the components of the parameters
To estimate the polynomial coefficients a _{ i }, Poisson term μ _{ j }, Fourier series coefficients C _{ j }, S _{ j } and phase p _{ j } by the leastsquares method, we need to know the frequencies ω _{ j } precisely. We employ Fourier analysis that is part of the improved version and the estimation procedure as follows:
In order to obtain the linear Poisson term μ _{ j }, we divide the data into subsets covering 300 yr and apply Fourier analysis to them. Then, by linear regression of the amplitudes as the first estimation, we determine whether the Poisson term is necessary or not. However, the disadvantage of this method is that any nonlinear term will influence the constant term. Then we apply the leastsquares method to fit the form (6).
5 Determination of lunar free librations
The amplitude and phases of free librations can be determined from the fitting process. The wobble mode
As for the latitude mode, thanks to the long term ephemeris series DE430 spanning 1100 yr, we can fit the small mode with an amplitude of
We need to point out that the free libration in longitude is combined with two forced libration terms (Eckhardt
1982
) arising from Venus at periods of 2.8923 yr and 2.8921 yr, and the mixed term has a period of 2.8917 yr (1056.207 d) with an amplitude of
In this case, we use Eckhardt’s theory to derive the two forced terms and then subtract them from the mixed term. By the leastsquares method, we can get the amplitude and phase of the longitudinal free libration with the assumed period. The obtained free libration in longitude has a period of 1056.16 d, an amplitude of
Table
Notes: The difference in phase between this paper and Rambaux and Williams ( 2011 ) of the latitude mode is because we used the sine function to fit but they used the cosine function. 

The ephemerides DE421 and DE430 include a fluid core with an oblate coremantle boundary. This will lead to the fourth free mode, i.e. free core nutation (FCN). The period of FCN, as given by Rambaux and Williams (
2011
), is 197 yr in DE421, but in DE430 the
As for P
_{1} and P
_{2}, we find the second largest amplitude term has a period of 2190 d (5.99 yr) with different amplitudes of
6 Discussion
In this paper, based on the most accurate ephemeris DE430, we have determined the periods, amplitudes and phases of three free libration modes by FFT, which are part of the improved version, and the leastsquares method. Comparisons with previous results are shown in Table
The amplitude of the free libration depends on excitation and damping processes. Free librations will eventually be damped to zero in the absence of excitation. The observational detection of free librations requires recent existence of excitation to counter damping. Without recent excitation, the free librations are therefore expected to be completely damped. Some possible excitation mechanisms have been proposed in the past, although they were not satisfactory. Passage through a resonance was proposed by Eckhardt ( 1993 ), and existence of a liquid core was proposed by Yoder ( 1981 ).
With the continuation of LLR, more and more distance data between the Moon and Earth have been accumulated. Moreover, during the extension of the Chinese Chang’e3 lander mission, lunar microwave ranging technology with submillicycle accuracy (Ping et al. 2017 ) was realized and implemented from 2015 in China. The new determination of free librations modes will enable investigation of their mechanisms.
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