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Free librations of the two-layer Moon and the possibilities of their detection
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Advances in Space Research 42 (2008) 1398–1404 www.elsevier.com/locate/asr
Free librations of the two-layer Moon and the possibilities of their detection N. Petrova a,b,*, A. Gusev a, N. Kawano b, H. Hanada b b
a Kazan State University, 18, Kremljevskaja str., Kazan 420008, Russia National Astronomical Observatory, 2-12 Hoshigaoka, Mizusawa, Iwate 023-0861, Japan
Received 21 January 2008; received in revised form 8 February 2008; accepted 8 February 2008
Abstract Based on the forthcoming second stage of the Japanese Lunar mission ILOM (2013), when an optical telescope will be set on the surface near one of the Lunar poles, the possibility to detect free Lunar modes (Chandler-like wobble and free-core nutation) is considered. The difference between the Lunar Eulerian and Chandler-like wobble is explained. The terms ‘‘arbitrary libration” and ‘‘free libration” are discussed. The geometrical and physical interpretations of the free polar motion over the Lunar surface are considered from the viewpoints of Lunar surface-based observations and the Lunar Navigation Almanac. The dependencies of the free libration period on the core’s radius, density, and ellipticity are modelled and discussed. Ó 2008 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Lunar free libration; Chandler-like wobble; Free Core Nutation; Polar motion; ILOM
1. Introduction Our current knowledge of the internal structure of the Moon is very limited and is mainly based on observations of its gravity field. Our intention is to detect fine variations in its rotation caused by a complex stratigraphy of the Lunar interior, using high-precision astronomical observations of the Lunar rotation in the forthcoming mission ILOM (Hanada et al., 2005; Noda et al., 2008; Kawano et al., 2003). In order to investigate the interior structure of the Moon, a preliminary study of its free librations must be conducted. On one hand, from libration observations, a considerable rotational dissipation was detected. As a result, the free modes should have long been damped. On the other hand, the same observations showed the presence of free modes in the present state of the Moon. This emphasizes the importance of the spin–orbit interaction. * Corresponding author. Address: Kazan State University, 18, Kremljevskaja str., Kazan 420008, Russia. E-mail addresses: [email protected], [email protected] (N. Petrova).
In particular, the resonant interaction with Venus, as well as a two- or three-layer model of a non-rigid Moon with a tidal dissipation or turbulent dissipation at the core–mantle boundary (CMB) needs to be developed. A model of a two-layer Moon is to become the first step in this direction. The Hamiltonian formalism provides a powerful tool for the study the rotation. The Lunar rotation is sensitive to its interior structure. Numerical models of the Lunar physical libration (LPhL) (Newhall and Williams, 1997; Williams et al., 2001), satisfying the modern Lunar laser data (LLR), necessarily include complex internal startigraphy of the Lunar body. To do this in the framework of an analytical theory is much more difficult. The main advantage of the analytical approach over the LPhL theory is an opportunity to separate the forced and free libration. Precision data of a laser location of the Moon give a good basis for determination of amplitudes and phases of the free libration. Yet the opportunity to study the rotation of a celestial body from its surface in the framework of the planned experiment, ILOM, opens even greater prospects in this direction.
0273-1177/$34.00 Ó 2008 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2008.02.017
N. Petrova et al. / Advances in Space Research 42 (2008) 1398–1404
2. The Hamiltonian of the lunar free rotation As a rule, theories for the rotation of rigid planets and large Moons are constructed on the basis of the theory of rotation of the Earth. The most accurate theory of rotation of the rigid Earth was developed by Kinoshita and Souchay. We also point out an extensive and comprehensive work by Mathews et al. (1991a,b) and the elegant analytical approaches of Dehant and Getino to the development of the rotation theory of the Earth’s model, composed of a rigid and non-symmetric mantle and a liquid outer/solid inner core. These methods may be successfully used for the Lunar physical libration theory, provided one takes into account the resonance character of the Lunar rotation. Let us consider the free rotation of the Moon assuming that it has a rigid mantle and an elliptical liquid core. In the absence of another body, the spin of the Moon will be a true Eulerian motion, provided that there are no external perturbations, and that the internal friction is neglected. According to Getino (1995), the core–mantle interaction is implemented through variation of the inertia tensor of the core (inertial deformation) in response to the inertial coupling, introduced by Kubo as a ‘tensional torque’ to explain the core–mantle interaction for the Earth. Getino has shown that in the case of a free motion of a rigid mantle–liquid core body, such an interaction can be described by introducing a suitable set of reference frames. Based on this approach, Getino and Ferrandiz (1997) and Gonzalez and Getino (1997) developed a canonical core–mantle rotation theory for the Earth. Petrova and Gusev (1999) later employed this technique to analyze the core–mantle interaction in the Moon. A comprehensive review of the Hamiltonian approach to rigid body dynamics and the Andoyer variables was carried out by Gurfil et al. (2007). We consider a Lunar model composed of a rigid mantle and a liquid core. Let OXYZ be a non-rotating inertial frame (Inertial System of Coordinates – ISC), Oxyz be a frame of principal axes of the total Moon rotating with an angular velocity x with respect to the ISC. We refer to the frame of the principal axes as the Dynamical System of Coordinates (DSC). Let Oxcyczc be a core-fixed frame rotating with angular velocity dx with respect to the mantle. We consider the dynamical figure of the Moon as a three-axial ellipsoid, so that the tensors of inertia of the total Moon, P, the mantle, Pm and the core, Pc, in the Oxyz frame are given by, respectively: 0
A
B P ¼ @0
0 0 Ac B Pc ¼ @ 0 0
0
0
C B 0 A; 0 C 0 Bc 0
0
1
0
Am
B Pm ¼ @ 0 1
C 0 A Cc
0
0 Bm 0
0
1
C 0 A; Cm ð1Þ
1399
where P = Pm + Pc. Regarding the dynamical figure of the core, the following assumptions can be made: Due to the interaction of subsystems, a core–mantle redistribution of ellipticity occurs. It is the exchange of the moments of inertia due to convective motion and friction between layers. In this case, the dynamical figure of the core is the same as the figure of the mantle. According to Clairaut’s theorem, the flattening of a uniformly rotating body is decreased to its center. In this case, the core’s ellipticity must be less than the mantle’s ellipticity. Ellipticity of the core may be greater than the ellipticity of the mantle, because the core of a planet gets frozen at early stages of the rotation evolution, when the rotation velocity is higher than the velocity in later periods. We have considered a model wherein a liquid homogeneous core possesses an axial symmetry (Ac = Bc) and has a dynamical ellipticity close to that of the mantle. With an appropriate definition of the core rotation (Gonzalez and Getino, 1997), the angular momentum vectors of the mantle and the core, Lm and Lc, can be expressed in the Oxyz frame as Lm = Pm x, Lc = Pc (x + dx), and the total angular momentum of the Moon becomes L = Lm + Lc = Px + Pcdx. The kinetic energy in this case can be written as 1 1 t 1 1 1 t 1 t 1 T ¼ Ltm P1 m Lm þ Lc Pc Lc ¼ ðL Lc Þ Pm ðL Lc Þ þ Lc Pc Lc 2 2 2 2 ð2Þ
The problem of construction of the theory consists of the choice of canonical angular variables q = (q1, q2, q3)T and qc = (qc1, qc2, qc3)T, which define position of DSC and DSCcore relative to ISC, and of the conjugate momenta p = (p1, p2, p3)T and pc = (pc1, pc2, pc3)T. On the basis of these variables, we constructed (Petrova and Gusev, 1999) the Hamiltonian of the free rotation H = T, using the standard technique of the Hamiltonian approach. As a result, we obtained 12 equations for 6 canonical angular variables and for 6 canonical momenta. At this stage, the analysis of the rotational motion of the Moon is identical to that of the Earth (Gonzalez and Getino, 1997). According to the Getino approach, the position of the DSC with respect to the ISC is described as shown in Fig. 1, the Andoyer planes being the planes perpendicular to the angular momenta of the total Moon L and the core Lc, respectively. The angles k, l, m, kc, lc, mc and the momenta K, M, N, Kc, Mc, Nc conjugate to the angles are the Andoyer canonical variables for the total Moon and the Lunar core, respectively.
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Fig. 1. Andoyer variables for a two-layer Moon.
3. Solution for the Lunar pole’s motion: free and arbitrary libration of the Moon We have constructed the Hamiltonian for free rotation of the Moon. However, the Eulerian motion of a body relative to the centre of mass in the absence of external forces, does not match the real Lunar rotation, which is described in the first approximation by Cassini’s laws. Physically, it means that in the absence of external forces, the Moon is not gravitationally locked relative to the Earth. As early as the beginning of the 20th century, Hayn showed that longitude libration is absent in the Eulerian motion. Therefore, considering an absolutely free rotation of the Moon is not meaningful. It is thus necessary at the least to describe the Lunar rotation in the gravity field of the Earth, i.e. to consider the resonant character of the Lunar rotation, including the potential of interaction of the Lunar body with the Earth (and with the Sun, in a more precise analysis). Because the mass of the Lunar core is less than 4% of the total mass of the Moon, we will not consider the gravitational interaction of the core with external bodies. In such an approach, the problem does not differ from the classical problem of a rigid body analysis. The potential of interaction of the Moon with the Earth and the Sun in a general form can be written down as (Petrova, 1996): X U ¼C Qijk ðS mn ; C mn ; tÞqi1 qj2 qk3 :
e ¼T U e . Fortunately, from a point of view in the form H of mathematics, these new terms do not break the structure of the Hamiltonian H for the Eulerian motion, because one can find similar terms in the expression for the kinetic energy (2), in which the only numerical coefficients to be e . The system of changed stem from the expression for U e corresponds Hamilton’s equations with the Hamiltonian H to the homogeneous system of equations of the perturbed motion. Practically, the described approach is somewhat similar to the Hamiltonian formulation used for modelling relative coorbital motion in the paper of Gurfil and Kasdin (2003), where the Hamiltonian is partitioned into a linear term and a high-order term. We use only the linear term at this stage. According to Getino’s approach, we have considered the simplest case – that of the polar motion. In this approximation, when the angle between the rotation axes of the mantle and the core is constant, and, as a consequence, the projection of the vector x on the minimal axis of inertia is equal to the mean diurnal rotation of the Moon, xC = const = X, we can obtain a solution for the two projections xA, xB of the angular velocity. These projections determine the equations of motion of the pole in the DSC system. Let us now construct a plane tangent to the celestial sphere at the pole of inertia of the Moon, C. In this plane, we construct a system of coordinates with axes (u, v), which are parallel to inertia axes A and B (Fig. 2). The instant position of the rotation pole relative to the inertia pole C (in angular scale) will be defined by the coordinates: u ¼ xxCA ¼ v1 ðc1 sin r1 t þ c2 cos r1 tÞ þ v2 ðc3 sin r2 t þ c4 cos r2 tÞ; v ¼ xxCB ¼ v3 ðc1 cos r1 t þ c2 sin r1 tÞ þ v4 ðc3 cos r2 t þ c4 sin r2 tÞ: ð3Þ
In (3), ci (i = 1; 2; 3; 4) are constants of integration, and the amplitudes vi,i = 1, 2, 3, 4 are functions of the main inertia moments of the total Moon, mantle and core (Getino, 1995; Gonzalez and Getino, 1997; Petrova and Gusev, 1999). The frequencies r1 and r2 are given by the expressions
hU i
Here, the coefficients Qijk are dependent on time and the Stockes’ coefficients Smn, Cmn (up to the fourth order), characterizing the Lunar dynamical figure. For the harmonics of the second order of the potential, we usually use the dimensionless moments of inertia c, b. Cassini’s motion of the Moon is described only by the part e ¼ U
3 X
Qðc; b; C mn ; S mn Þq2k ;
C D2=8.19
v (B) F1
D1=3.31 F2
u (A)
k¼1
where the numerical coefficients Qðc; b; C mn ; S mn Þ are not depended on time. We can now consider the Hamiltonian
Fig. 2. The motion of the rotation pole relative to inertia axis pole due to CW and FCN.
N. Petrova et al. / Advances in Space Research 42 (2008) 1398–1404
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ðC AÞðC BÞ A pffiffiffiffiffiffiffiffi ¼X j r1 ¼ X jab ! CW Am AB Am A C c Ac C c B c r2 ¼ X 1 þ þ 2Am Ac Bc A ¼ X 1 þ ðeca þ ecb Þ ! FCN 2Am
Table 1 Numerical values of the Lunar parameters used for modelling
ð4Þ
Unlike Eulerian rotations, where the frequency r1 is usually referred to as a Chandler-like wobble, the factor j = 3.99516 4 appears under a square root. It comes into e . Such a libration the solution from the potential part U cannot be called ‘‘free” libration, because it occurs due to the orbital motion of the Moon, i.e. under the action of an external force. Hayn (1923) has obtained harmonics with arbitrary constants, and realized the physical consequence of the expressions, consequently naming them not ‘‘free libration” (Freie Libration), but rather ‘‘arbitrary libration” (Willku¨rliche Libration). Khabibullin (1968, 1969) also pointed out this feature of the solution of the homogeneous system of libration equations. The arbitrary libration is a forced motion under the action of the linear part of an external torque. The frequency r2 appears in the solution due to the core and represents the so-called Free Core Nutation (FCN). The minus sign before the expression means that the oscillation occurs in a retrograde direction relative to the orbital motion. Getino et al. have called this frequency Retrograde Fluid Core Nutation. The rotation axis of the core and the mantle are not aligned, and thus the FCN takes place. Displacement of the axes may be explained by various reasons. One of them is the fact that the mantle is subjected to the Earth attraction, but the core is rotating freely due to its small mass. To each of the obtained frequencies, the respective period can be calculated: P rot Am P rot Am 1 P CW ¼ pffiffiffipffiffiffiffiffiffi ffi pffiffiffiffiffiffi ¼ P Eulerian 2 A 2 ab A j ab
ð5Þ
Prot = 27.3 days is the sidereal period of the Lunar rotation and is equal to 1 Lunar sidereal month. The values used for numerical estimates are presented in Table 1. We obtained Pcw = 74.07 years (Petrova and Gusev, p 2001; ffiffiffi Barkin et al., 2006). This value is two times smaller ð jÞ than the Eulerian period. This fact shows a distinction between the arbitrary and Eulerian polar motion. For the second frequency, r2, the respective period, P2, is close to the Lunar month. In order to determine the difference between the Lunar month and P2, the expression for r2 is usually rewritten into another form. For the Moon, we have obtained the following expression: 1 1 þ r2 ¼ ð6Þ 1 month P FCN and, consequently, the FCN-period is expressed as
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Mean radius of the Moon Mean angular velocity
RL X
Dynamical ellipticities
b 104 = (C A)/B c 104 = (B A)/C a 104 = (b c)/(1 c) cþ1 ea ¼ CA A ¼ b 1cb b bc CB eb ¼ B ¼ 1þc a
Geometrical ellipticity Normalized moment of inertia Mantle density Core density
C/MR2
1738.09 km 3.6601 102 (rev/day) 6.31486 ± 0.0009 2.27871 ± 0.0003 4,030707 6.31 104 4,03 104 4 104 0.3932 ± 0.0002
qm qc (eutectic Fe–FeS) qc (liquid Fe) qc (solid Fe)
3,269 gm cm3 5.3 gm cm3 7.0 gm cm3 7.7 gm cm3
ac a
P rot Am ! for a tri-axial core; 2ðeca þ ecb Þ A P rot Am ! for a core with axial symmetry ¼ ec A
P FCN ¼ P FCN
ð7Þ
Our estimates are PFCN = 144–186 years. Such large values (relative to the Earth and Mars) of the Lunar free periods are explained by the small size of the core and by the slow Lunar rotation (Barkin et al., 2006). 4. Geometrical and physical interpretation of the periods Eq. (3) may be rewritten into the more common form u ¼ D1 cosðr1 t þ dÞ þ F 1 cosðr2 t þ f Þ v ¼ D2 sinðr1 t þ dÞ þ F 2 sinðr2 t þ f Þ where the amplitudes D1, F1, D2, F2 and phases d and f contain undefined constants of integration, whose values can be determined from observations. These equations show that the polar motion on a Lunar surface may consist of two motions over ellipses (Fig. 2). The pole moves in a retrograde direction over a small ellipse with semi-axes F1 and F2 and a period that equals approximately to the Lunar month. Its centre moves along the ellipse with semi-axes D1 and D2 in the prograde direction with a period of 74 years. In reality, the axis of inertia moves around the mean Lunar rotation axis (Fig. 3), because the inertia axes are rigidly connected to the Lunar rigid body. In the inertial frame, this motion is imposed on the retrograde precession motion of the vector of the mean angu inclined to the ecliptic pole by I = 1.53°. lar velocity, x, The non-coincidence of the axes of inertia and rotation is the reason for the Chandler wobble. Calame (1977) was the first to detect the two larger free libration modes from Lunar laser ranging. Based on 35year series of LLR-measurements, the amplitudes and phases of the Lunar free libration were detected with good accuracy (Williams et al., 2001). According to these data, the amplitudes D1 = 3.3100 , D2 = 8.1900 (Newhall and
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z (C)
y (B)
x (A) Fig. 3. The motion of the inertia axis C relative to the rotation axis due to the Chandler wobble. The ellipse is of wave form due to the FCN.
Williams, 1997), and the period is 74.63 years. This value is close to the one theoretically predicted. This fact once again confirms that the Moon may be well described by the rigid body model with very small elasticity. For the Earth, the Chandler wobble period (433 days) is significantly larger than the Eulerian period (355 days). The full explanation for this discrepancy involves the fluid nature of the Earth’s core and oceans. As for the Free Core Nutation, its appearance in the polar motion is due to a non-coincidence of the rotation axes of the core and total mantle. This mode lies in the Lunar monthly (‘‘Lunar diurnal”) band, but it is a retrograde mode, thus yielding a period smaller than the one Lunar month duration. It will be difficult to detect the FCN, because it blends with more strong resonant harmonics of the forced libration. Moreover the estimated magnitude of PFCN is large and, consequently, the difference of 1 month in (6) is insignificant, i.e. the contribution of the Lunar core is small. Nevertheless, due to the fact that the FCN is in the monthly band, this mode may be of particular interest because it could influence some monthly tides as well as the forced amplitudes–resonance amplification. Certainly, as a whole, the FCN-problem is similar to the terrestrial one, but because of the small sizes of the Lunar core all its effects are much less significant. The forthcoming Japanese experiment ILOM, in which it is planned to establish an optical telescope on one of the poles of the Moon comprises the compact PZT telescope, which will observe the motion of the stars around the pole in order to measure the physical librations of the Moon. It will be fixed to the celestial pole, and equipped with a CCD camera with 4000 4000 pixels to determine the instantaneous position of several tens of stars, accurate to the order of 1 milli-arc-second (1 mas). The first stage of the Japanese mission SELENE (Kaguya) was launched in September, 2007. At the present time it successfully functions. One of the major results of this experiment will be the improvement of a model of the Lunar gravitational field and, in particular, specification of the harmonics of the low order (Goossens et al., 2008; Namiki et al., 2008). All this will allow for signifi-
cantly improving the accuracy of LPhL-theory and hence will raise chances to detect the more fine effects in the Lunar rotation, such as the FCN. The polar telescope (Fig. 3) will show the motion of the Lunar main inertia axis relative to the stars, i.e. relative to the inertial frame. Moreover, from the Lunar surface, it will be possible to detect FCN by those methods that were successfully used for the definition of the FCN of the Earth. In a sense, we will see an opposite picture of the polar motion in the Lunar body if we compare the Moon with the Earth. The amplitude of the Eulerian motions of the Earth’s pole can exceed 0.500 . It is much more than the amplitude of the forced motion, caused by the luni-solar perturbations. That is why only the Eulerian motion is usually taken into consideration when the astronomical observations are being processed. For the Moon, the amplitude of the pole’s forced librations is big and can exceed 20 . The free libration’s amplitude is essentially smaller and does not exceed 900 . Moreover, the free libration’s periods are tens and hundreds of years. This fact should be taken into consideration when the Lunar Navigational Almanac will be developed for future observations at the Lunar surface. When astrometrical observations are processed, the forced motion of the Lunar pole should be taken in the first order; but it is possible not to consider the free motion during 3–5 years. In spite of these differences, the inclusion of the pole’s motion into processing of astrometrical observations is almost the same for the Moon and for the Earth (Kulikov and Gurevich, 1972). Such a fine effect as the FCN has not been observed from the Earth yet. One have to take into account that this fact does not indicate the absence of these phenomena, but it is an evidence of imperfection of the existing methods of processing the observational data. This imperfection consists of the following. Different interpretations of the results are possible, because only those small components of the Moon’s rotation are determined by calculations whose existence was assumed beforehand. For example, the forced libration harmonics or the Chandler-like wobble. Those components, which have not been modeled a priori, cannot be detected. Their existence may be manifested indirectly through decreasing the observational noise. However, it is necessary to point out that if the FCN-mode is detected from observations, then the existence of a Lunar liquid core will be confirmed directly, because only for a planet containing a liquid core, the FCN may be observed. From the LLR, a significant dissipation of the Lunar rotation is detected (Yoder, 1981; Williams et al., 2001). Dissipation is manifested as a small 0.2600 advance in the Lunar rotation axis relative to Cassini’s position. Dissipation from flexing and fluid core interactions should cause the Lunar free polar motion to damp with geologically short time scales. Thus, the detected finite amplitudes imply recent or active stimulation. It has been suggested by Yoder (1981) that this mode may be stimulated by eddies
N. Petrova et al. / Advances in Space Research 42 (2008) 1398–1404
at the solid-mantle/liquid-core boundary. Both damping and active excitation may be observable in the future, and active excitation should cause temporal irregularities in this mode. The free rotational modes for a fluid core and any core–mantle boundary flattening would give rise to a resonant FCN frequency.
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CORE
5. Modeling the free periods of the Moon MANTLE
6. Conclusions In this paper, the polar motion for a rigid mantle–liquid core lunar model has been considered. Two modes in its polar motion can be observed in this case. The Chandlerlike mode is already detected from Laser ranging observations (Newhall and Williams, 1997). It manifests itself as an
74,20
Pcw (year)
74,00
4,5 gm/cm 3
73,80
5 5,5
73,60
6 73,40
7
73,20 200
250
300
350
400
450
500
550
600
Rc (km) 144,60 144,40 144,20
PFCN (year)
The modeling was carried out with an intention to detect a dependence of the free libration periods on various parameters characterizing a Lunar liquid core. The parameters of the core – size, ellipticity, densities, and the state of aggregation – were attributed values obtained from the LLR-analyses (Table 1) (Williams et al., 2001). A model of the Moon composed of a rigid mantle and a liquid core of various density values was considered (Fig. 4). The chemical composition of the core was set to vary from a eutectic composition Fe–FeS (with various contents of sulphur) to a purely iron core. The density of the eutectic composition varied from 4 gm/cm3 (a high content of S) up to 7 gm/cm3 (100% Fe). According to Williams et al. (2001), the density of 5.3 gm/cm3 corresponds to 25% wt S. Constraints on the radius of the core at various densities were derived from the fact that the mass of the core should not exceed 4–5% of the Lunar mass. To that end, the normalized moment of inertia C/MR2 has been also calculated and compared to the observed one. As a result, the range from 200 to 600 km was accepted. The core possesses an axial symmetry and has ellipticity, which can be calculated in two ways – either through the mean ellipticity of the mantle’s equator’s, or as a geometrical ellipticity estimated from the LLR-analyses (Dickey et al., 1994). Calculations were made for two values of the core’s ellipticity: ec1 = (ea + eb)/2 = 5.2 104 (the mean dynamical ellipticity of the mantle’s equator), and ec2 ¼ ac ¼ 4 104 (the value of geometrical ellipticity a estimated from LLR-analyses, Dickey et al., 1994). Here a, c are the core’s maximal and minimal semi-axes. In the approximation of a homogeneous core, the dynamical and geometrical ellipticities are equal. The Chandler-like wobble period is independent of the core’s ellipticity, but for the FCN we have obtained for these two ellipticities quite different values: 144 years for ec1 and 186 years for ec2. The dependence of the free periods on the core’s radius for a variety of densities is presented in Fig. 4.
4.5 gm/cm 3
144,00
5 143,80
5,5 143,60
6 143,40
7
143,20 143,00 200
250
300
350
400
450
500
550
600
Rc (km)
Fig. 4. Two-layer model of the Moon. Dependence of the PCW (top) and PFCN (bottom) on the core’s radius for various values of density.
elliptical path of the principal body axis with amplitudes of 8.1900 and 3.3100 . The major axis of the displacement is parallel to the lunar principal B-moment of inertia. The wobble period is 74.63 years. The second mode, Free Core Nutation, can be considered as an additional retrograde elliptical motion over the Chandler’s ellipse with amplitude much smaller than the detected Chandler-like amplitudes, while the period is close to the Lunar sidereal month. According to the classical determination of the FCN-period (Eq. (6)), its magnitude may be estimated by Eq. (7). The forthcoming Japanese experiment ILOM (2013) offers a greater opportunity to detect this mode due to the expected high-accuracy of the observations, and because it will permit the use of methods developed for detection of the terrestrial FCN. Due to the fact that the FCN is in the monthly band, this mode may be of particular interest because it could influence some monthly tides as well as the forced amplitudes–resonance amplification. An analysis of results of the modelling has allowed us to draw the following conclusions:
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1. The dependence on the core’s radius is very weak for both periods (PCW and PFCN), when the radius is set to vary within the range of 300–600 km, the periods are changed by less than 1%. 2. Both periods depend very weakly on the core’s density. Only for radii greater than 400 km, the difference in density is observed. The difference in the periods between the eutectic composition with the density 5.5 gm/cm3 and the pure iron core (7 gm/cm3) is less than 0.04%. 3. The FCN-period is very sensitive to the core’s ellipticity. This property can be used to impose an additional constraint on the core’s parameters, if the expected observation data allows detecting the FCN-mode in the polar motion. The expected amplitudes of the free core’s libration lie within the range of 1–3 ms of an arc (Yoder, 1981). Acknowledgements The authors express their gratitude to Michael Efroimsky, Pini Gurfil and Frank Lemoine for the fruitful discussion and useful advices, which have essentially improved the content of the paper. The research described in this paper was supported by the Russian–Japanese Grant RFFIJSPS N 07-02-91212 (2007–2009). References Barkin, Yu., Gusev, A., Petrova, N. Study of spin–orbit and inner dynamics of the Moon: Lunar mission applications. Adv. Space Res. 37, 72–79, 2006. Calame, O. Free librations of the Moon from Lunar laser ranging, in: Mulholland, J.D. (Ed.), Scientific Applications of Lunar Laser Ranging. Reidel, Dordrecht/Boston, pp. 53–63, 1977. Dickey, J.O., Bender, P.L., Newhall, X.X., et al. Lunar laser ranging: a continuing legacy of the Apollo program. Science 265, 482, 1994. Getino, J. An interpretation of the core–mantle interaction problem. Geophys. J. Int. 120, 693, 1995. Getino, J., Ferrandiz, J.M. A Hamiltonian approach to dissipative phenomena between the Earth’s mantle and core, and effects on free nutations. Geophys. J. Int. 130, 326–334, 1997. Gonzalez, A.-B., Getino, J. The rotation of a non-rigid, non-symmetrical Earth I: free nutations. Celestial Mech. Dyn. Astron. 68, 139–149, 1997. Goossens, S., Matsumoto, K., Liu, Q., Noda, H., Namiki, N., Iwata, T. On the processing of tracking data from the SELENE satellites for orbit and lunar gravity field determination. Abstracts of the 2-nd
KAGUYA (SELENE) Working Meeting. January 15–18, Tokyo, Japan, 2008. Gurfil, P., Elipe, A., Efroimsky, M., Tangren, W. The Serret–Andoyer formalism in rigid-body dynamics: I. Symmetries and perturbations. Reg. Chaotic Dyn. 12 (4), 389–425, 2007. Gurfil, P., Kasdin, N.J. Canonical modelling of coorbital motion in Hill’s problem using epicyclic orbital elements. A&A 409 (3), 1135–1140, 2003. Hanada, H., Heki, K., Kawano, N., et al. Application of a PZT telescope to in situ lunar orientation measurement (ILOM), in: Proc. Int. Ass. Geod. IAG GA, Sapporo, Japan, June 30–July 11, 2003. Springer, pp. 163–168, 2005. Hayn, F. DieRotation des Mondes. Enc. d. Math. Wiss. Bd. VI, 2 Teil, Heft 8, 1923. Kawano, N., Hanada, H., Tsubokawa, T., Iwata, T. Research on Lunar interior by Japanese explorer SELENE, in: Proc. Intl. Conf. ‘‘New Geometry of Nature”, August 25–September 5, 2003. Kazan University Press, vol. 3, pp. 114–119, 2003. Khabibullin, Sh.T. System of selenographic coordinates, precession and nutation of the Moon’s rotation axis. SvA (AJ) 12 (3), 526–534, 1968. Khabibullin, Sh.T., Chikanov, Yu.A. Arbitrary libration and Eulerian motion of the Lunar poles. Trudy Kazanskoj Observation, pp. 49–60, 1969 (in Russian). Kulikov, K.A., Gurevich, V. Lunar astrometry, M.: Nauka, p. 392, 1972 (in Russian). Mathews, P.M., Buffett, B.A., Herring, T.A., Shapiro, I.I. Forced nutations of the Earth: Influence of inner core dynamics: I. Theory. J. Geophys. Res. 96 (B5), 8219–8242, 1991a. Mathews, P.M., Buffett, B.A., Herring, T.A., Shapiro, I.I. Forced nutations of the Earth: Influence of inner core dynamics: II. Numerical results and comparisons. J. Geophys. Res. 96 (B5), 8243– 8257, 1991b. Namiki, N., Iwata, T., Matsumoto, K., Hanada, H., Noda, H., Ogawa, M., Kawano, N., et al. ‘‘RSAT”. Initial results of gravity experiment by four-way Doppler measurement. Abstracts of the 2-nd KAGUYA (SELENE) Working Meeting, January 15–18, Tokyo, Japan, 2008. Noda, H., Heki, K., Hanada, H. In situ Lunar orientation measurement (ILOM): simulation of observation. Adv. Space Res. 42, 358–362, 2008. Newhall, X.X., Williams, J.G. Estimation of the Lunar physical librations. Celestial Mech. Dyn. Astron. 66, 21–30, 1997. Petrova, N. Analytical extension of Lunar libration tables. EM&P 73 (1), 71–99, 1996. Petrova, N., Gusev, A. Free core nutation of the Moon, in: Proc. Intl. Conf. ‘‘Geom. Phys. IV”, Kazan. Kazan University Press, pp. 276– 282, 1999. Petrova, N., Gusev, A. New trends in the development of the Lunar physical libration theory. Celestial Mech. Dyn. Astron. 80 (3/4), 215– 225, 2001. Williams, J.G., Boggs, D., Yoder, Ch., Ratcliff, J., Dickey, J. Lunar rotational dissipation in solid body and molten core. J. Geophys. Res. 106, No. E11, pp. 27, 933–27, 968, 2001. Yoder, C.F. The free librations of a dissipative Moon. PTRSL 303, 327– 338, 1981.
Advances in Space Research 42 (2008) 1398–1404 www.elsevier.com/locate/asr
Free librations of the two-layer Moon and the possibilities of their detection N. Petrova a,b,*, A. Gusev a, N. Kawano b, H. Hanada b b
a Kazan State University, 18, Kremljevskaja str., Kazan 420008, Russia National Astronomical Observatory, 2-12 Hoshigaoka, Mizusawa, Iwate 023-0861, Japan
Received 21 January 2008; received in revised form 8 February 2008; accepted 8 February 2008
Abstract Based on the forthcoming second stage of the Japanese Lunar mission ILOM (2013), when an optical telescope will be set on the surface near one of the Lunar poles, the possibility to detect free Lunar modes (Chandler-like wobble and free-core nutation) is considered. The difference between the Lunar Eulerian and Chandler-like wobble is explained. The terms ‘‘arbitrary libration” and ‘‘free libration” are discussed. The geometrical and physical interpretations of the free polar motion over the Lunar surface are considered from the viewpoints of Lunar surface-based observations and the Lunar Navigation Almanac. The dependencies of the free libration period on the core’s radius, density, and ellipticity are modelled and discussed. Ó 2008 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Lunar free libration; Chandler-like wobble; Free Core Nutation; Polar motion; ILOM
1. Introduction Our current knowledge of the internal structure of the Moon is very limited and is mainly based on observations of its gravity field. Our intention is to detect fine variations in its rotation caused by a complex stratigraphy of the Lunar interior, using high-precision astronomical observations of the Lunar rotation in the forthcoming mission ILOM (Hanada et al., 2005; Noda et al., 2008; Kawano et al., 2003). In order to investigate the interior structure of the Moon, a preliminary study of its free librations must be conducted. On one hand, from libration observations, a considerable rotational dissipation was detected. As a result, the free modes should have long been damped. On the other hand, the same observations showed the presence of free modes in the present state of the Moon. This emphasizes the importance of the spin–orbit interaction. * Corresponding author. Address: Kazan State University, 18, Kremljevskaja str., Kazan 420008, Russia. E-mail addresses: [email protected], [email protected] (N. Petrova).
In particular, the resonant interaction with Venus, as well as a two- or three-layer model of a non-rigid Moon with a tidal dissipation or turbulent dissipation at the core–mantle boundary (CMB) needs to be developed. A model of a two-layer Moon is to become the first step in this direction. The Hamiltonian formalism provides a powerful tool for the study the rotation. The Lunar rotation is sensitive to its interior structure. Numerical models of the Lunar physical libration (LPhL) (Newhall and Williams, 1997; Williams et al., 2001), satisfying the modern Lunar laser data (LLR), necessarily include complex internal startigraphy of the Lunar body. To do this in the framework of an analytical theory is much more difficult. The main advantage of the analytical approach over the LPhL theory is an opportunity to separate the forced and free libration. Precision data of a laser location of the Moon give a good basis for determination of amplitudes and phases of the free libration. Yet the opportunity to study the rotation of a celestial body from its surface in the framework of the planned experiment, ILOM, opens even greater prospects in this direction.
0273-1177/$34.00 Ó 2008 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2008.02.017
N. Petrova et al. / Advances in Space Research 42 (2008) 1398–1404
2. The Hamiltonian of the lunar free rotation As a rule, theories for the rotation of rigid planets and large Moons are constructed on the basis of the theory of rotation of the Earth. The most accurate theory of rotation of the rigid Earth was developed by Kinoshita and Souchay. We also point out an extensive and comprehensive work by Mathews et al. (1991a,b) and the elegant analytical approaches of Dehant and Getino to the development of the rotation theory of the Earth’s model, composed of a rigid and non-symmetric mantle and a liquid outer/solid inner core. These methods may be successfully used for the Lunar physical libration theory, provided one takes into account the resonance character of the Lunar rotation. Let us consider the free rotation of the Moon assuming that it has a rigid mantle and an elliptical liquid core. In the absence of another body, the spin of the Moon will be a true Eulerian motion, provided that there are no external perturbations, and that the internal friction is neglected. According to Getino (1995), the core–mantle interaction is implemented through variation of the inertia tensor of the core (inertial deformation) in response to the inertial coupling, introduced by Kubo as a ‘tensional torque’ to explain the core–mantle interaction for the Earth. Getino has shown that in the case of a free motion of a rigid mantle–liquid core body, such an interaction can be described by introducing a suitable set of reference frames. Based on this approach, Getino and Ferrandiz (1997) and Gonzalez and Getino (1997) developed a canonical core–mantle rotation theory for the Earth. Petrova and Gusev (1999) later employed this technique to analyze the core–mantle interaction in the Moon. A comprehensive review of the Hamiltonian approach to rigid body dynamics and the Andoyer variables was carried out by Gurfil et al. (2007). We consider a Lunar model composed of a rigid mantle and a liquid core. Let OXYZ be a non-rotating inertial frame (Inertial System of Coordinates – ISC), Oxyz be a frame of principal axes of the total Moon rotating with an angular velocity x with respect to the ISC. We refer to the frame of the principal axes as the Dynamical System of Coordinates (DSC). Let Oxcyczc be a core-fixed frame rotating with angular velocity dx with respect to the mantle. We consider the dynamical figure of the Moon as a three-axial ellipsoid, so that the tensors of inertia of the total Moon, P, the mantle, Pm and the core, Pc, in the Oxyz frame are given by, respectively: 0
A
B P ¼ @0
0 0 Ac B Pc ¼ @ 0 0
0
0
C B 0 A; 0 C 0 Bc 0
0
1
0
Am
B Pm ¼ @ 0 1
C 0 A Cc
0
0 Bm 0
0
1
C 0 A; Cm ð1Þ
1399
where P = Pm + Pc. Regarding the dynamical figure of the core, the following assumptions can be made: Due to the interaction of subsystems, a core–mantle redistribution of ellipticity occurs. It is the exchange of the moments of inertia due to convective motion and friction between layers. In this case, the dynamical figure of the core is the same as the figure of the mantle. According to Clairaut’s theorem, the flattening of a uniformly rotating body is decreased to its center. In this case, the core’s ellipticity must be less than the mantle’s ellipticity. Ellipticity of the core may be greater than the ellipticity of the mantle, because the core of a planet gets frozen at early stages of the rotation evolution, when the rotation velocity is higher than the velocity in later periods. We have considered a model wherein a liquid homogeneous core possesses an axial symmetry (Ac = Bc) and has a dynamical ellipticity close to that of the mantle. With an appropriate definition of the core rotation (Gonzalez and Getino, 1997), the angular momentum vectors of the mantle and the core, Lm and Lc, can be expressed in the Oxyz frame as Lm = Pm x, Lc = Pc (x + dx), and the total angular momentum of the Moon becomes L = Lm + Lc = Px + Pcdx. The kinetic energy in this case can be written as 1 1 t 1 1 1 t 1 t 1 T ¼ Ltm P1 m Lm þ Lc Pc Lc ¼ ðL Lc Þ Pm ðL Lc Þ þ Lc Pc Lc 2 2 2 2 ð2Þ
The problem of construction of the theory consists of the choice of canonical angular variables q = (q1, q2, q3)T and qc = (qc1, qc2, qc3)T, which define position of DSC and DSCcore relative to ISC, and of the conjugate momenta p = (p1, p2, p3)T and pc = (pc1, pc2, pc3)T. On the basis of these variables, we constructed (Petrova and Gusev, 1999) the Hamiltonian of the free rotation H = T, using the standard technique of the Hamiltonian approach. As a result, we obtained 12 equations for 6 canonical angular variables and for 6 canonical momenta. At this stage, the analysis of the rotational motion of the Moon is identical to that of the Earth (Gonzalez and Getino, 1997). According to the Getino approach, the position of the DSC with respect to the ISC is described as shown in Fig. 1, the Andoyer planes being the planes perpendicular to the angular momenta of the total Moon L and the core Lc, respectively. The angles k, l, m, kc, lc, mc and the momenta K, M, N, Kc, Mc, Nc conjugate to the angles are the Andoyer canonical variables for the total Moon and the Lunar core, respectively.
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Fig. 1. Andoyer variables for a two-layer Moon.
3. Solution for the Lunar pole’s motion: free and arbitrary libration of the Moon We have constructed the Hamiltonian for free rotation of the Moon. However, the Eulerian motion of a body relative to the centre of mass in the absence of external forces, does not match the real Lunar rotation, which is described in the first approximation by Cassini’s laws. Physically, it means that in the absence of external forces, the Moon is not gravitationally locked relative to the Earth. As early as the beginning of the 20th century, Hayn showed that longitude libration is absent in the Eulerian motion. Therefore, considering an absolutely free rotation of the Moon is not meaningful. It is thus necessary at the least to describe the Lunar rotation in the gravity field of the Earth, i.e. to consider the resonant character of the Lunar rotation, including the potential of interaction of the Lunar body with the Earth (and with the Sun, in a more precise analysis). Because the mass of the Lunar core is less than 4% of the total mass of the Moon, we will not consider the gravitational interaction of the core with external bodies. In such an approach, the problem does not differ from the classical problem of a rigid body analysis. The potential of interaction of the Moon with the Earth and the Sun in a general form can be written down as (Petrova, 1996): X U ¼C Qijk ðS mn ; C mn ; tÞqi1 qj2 qk3 :
e ¼T U e . Fortunately, from a point of view in the form H of mathematics, these new terms do not break the structure of the Hamiltonian H for the Eulerian motion, because one can find similar terms in the expression for the kinetic energy (2), in which the only numerical coefficients to be e . The system of changed stem from the expression for U e corresponds Hamilton’s equations with the Hamiltonian H to the homogeneous system of equations of the perturbed motion. Practically, the described approach is somewhat similar to the Hamiltonian formulation used for modelling relative coorbital motion in the paper of Gurfil and Kasdin (2003), where the Hamiltonian is partitioned into a linear term and a high-order term. We use only the linear term at this stage. According to Getino’s approach, we have considered the simplest case – that of the polar motion. In this approximation, when the angle between the rotation axes of the mantle and the core is constant, and, as a consequence, the projection of the vector x on the minimal axis of inertia is equal to the mean diurnal rotation of the Moon, xC = const = X, we can obtain a solution for the two projections xA, xB of the angular velocity. These projections determine the equations of motion of the pole in the DSC system. Let us now construct a plane tangent to the celestial sphere at the pole of inertia of the Moon, C. In this plane, we construct a system of coordinates with axes (u, v), which are parallel to inertia axes A and B (Fig. 2). The instant position of the rotation pole relative to the inertia pole C (in angular scale) will be defined by the coordinates: u ¼ xxCA ¼ v1 ðc1 sin r1 t þ c2 cos r1 tÞ þ v2 ðc3 sin r2 t þ c4 cos r2 tÞ; v ¼ xxCB ¼ v3 ðc1 cos r1 t þ c2 sin r1 tÞ þ v4 ðc3 cos r2 t þ c4 sin r2 tÞ: ð3Þ
In (3), ci (i = 1; 2; 3; 4) are constants of integration, and the amplitudes vi,i = 1, 2, 3, 4 are functions of the main inertia moments of the total Moon, mantle and core (Getino, 1995; Gonzalez and Getino, 1997; Petrova and Gusev, 1999). The frequencies r1 and r2 are given by the expressions
hU i
Here, the coefficients Qijk are dependent on time and the Stockes’ coefficients Smn, Cmn (up to the fourth order), characterizing the Lunar dynamical figure. For the harmonics of the second order of the potential, we usually use the dimensionless moments of inertia c, b. Cassini’s motion of the Moon is described only by the part e ¼ U
3 X
Qðc; b; C mn ; S mn Þq2k ;
C D2=8.19
v (B) F1
D1=3.31 F2
u (A)
k¼1
where the numerical coefficients Qðc; b; C mn ; S mn Þ are not depended on time. We can now consider the Hamiltonian
Fig. 2. The motion of the rotation pole relative to inertia axis pole due to CW and FCN.
N. Petrova et al. / Advances in Space Research 42 (2008) 1398–1404
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ðC AÞðC BÞ A pffiffiffiffiffiffiffiffi ¼X j r1 ¼ X jab ! CW Am AB Am A C c Ac C c B c r2 ¼ X 1 þ þ 2Am Ac Bc A ¼ X 1 þ ðeca þ ecb Þ ! FCN 2Am
Table 1 Numerical values of the Lunar parameters used for modelling
ð4Þ
Unlike Eulerian rotations, where the frequency r1 is usually referred to as a Chandler-like wobble, the factor j = 3.99516 4 appears under a square root. It comes into e . Such a libration the solution from the potential part U cannot be called ‘‘free” libration, because it occurs due to the orbital motion of the Moon, i.e. under the action of an external force. Hayn (1923) has obtained harmonics with arbitrary constants, and realized the physical consequence of the expressions, consequently naming them not ‘‘free libration” (Freie Libration), but rather ‘‘arbitrary libration” (Willku¨rliche Libration). Khabibullin (1968, 1969) also pointed out this feature of the solution of the homogeneous system of libration equations. The arbitrary libration is a forced motion under the action of the linear part of an external torque. The frequency r2 appears in the solution due to the core and represents the so-called Free Core Nutation (FCN). The minus sign before the expression means that the oscillation occurs in a retrograde direction relative to the orbital motion. Getino et al. have called this frequency Retrograde Fluid Core Nutation. The rotation axis of the core and the mantle are not aligned, and thus the FCN takes place. Displacement of the axes may be explained by various reasons. One of them is the fact that the mantle is subjected to the Earth attraction, but the core is rotating freely due to its small mass. To each of the obtained frequencies, the respective period can be calculated: P rot Am P rot Am 1 P CW ¼ pffiffiffipffiffiffiffiffiffi ffi pffiffiffiffiffiffi ¼ P Eulerian 2 A 2 ab A j ab
ð5Þ
Prot = 27.3 days is the sidereal period of the Lunar rotation and is equal to 1 Lunar sidereal month. The values used for numerical estimates are presented in Table 1. We obtained Pcw = 74.07 years (Petrova and Gusev, p 2001; ffiffiffi Barkin et al., 2006). This value is two times smaller ð jÞ than the Eulerian period. This fact shows a distinction between the arbitrary and Eulerian polar motion. For the second frequency, r2, the respective period, P2, is close to the Lunar month. In order to determine the difference between the Lunar month and P2, the expression for r2 is usually rewritten into another form. For the Moon, we have obtained the following expression: 1 1 þ r2 ¼ ð6Þ 1 month P FCN and, consequently, the FCN-period is expressed as
1401
Mean radius of the Moon Mean angular velocity
RL X
Dynamical ellipticities
b 104 = (C A)/B c 104 = (B A)/C a 104 = (b c)/(1 c) cþ1 ea ¼ CA A ¼ b 1cb b bc CB eb ¼ B ¼ 1þc a
Geometrical ellipticity Normalized moment of inertia Mantle density Core density
C/MR2
1738.09 km 3.6601 102 (rev/day) 6.31486 ± 0.0009 2.27871 ± 0.0003 4,030707 6.31 104 4,03 104 4 104 0.3932 ± 0.0002
qm qc (eutectic Fe–FeS) qc (liquid Fe) qc (solid Fe)
3,269 gm cm3 5.3 gm cm3 7.0 gm cm3 7.7 gm cm3
ac a
P rot Am ! for a tri-axial core; 2ðeca þ ecb Þ A P rot Am ! for a core with axial symmetry ¼ ec A
P FCN ¼ P FCN
ð7Þ
Our estimates are PFCN = 144–186 years. Such large values (relative to the Earth and Mars) of the Lunar free periods are explained by the small size of the core and by the slow Lunar rotation (Barkin et al., 2006). 4. Geometrical and physical interpretation of the periods Eq. (3) may be rewritten into the more common form u ¼ D1 cosðr1 t þ dÞ þ F 1 cosðr2 t þ f Þ v ¼ D2 sinðr1 t þ dÞ þ F 2 sinðr2 t þ f Þ where the amplitudes D1, F1, D2, F2 and phases d and f contain undefined constants of integration, whose values can be determined from observations. These equations show that the polar motion on a Lunar surface may consist of two motions over ellipses (Fig. 2). The pole moves in a retrograde direction over a small ellipse with semi-axes F1 and F2 and a period that equals approximately to the Lunar month. Its centre moves along the ellipse with semi-axes D1 and D2 in the prograde direction with a period of 74 years. In reality, the axis of inertia moves around the mean Lunar rotation axis (Fig. 3), because the inertia axes are rigidly connected to the Lunar rigid body. In the inertial frame, this motion is imposed on the retrograde precession motion of the vector of the mean angu inclined to the ecliptic pole by I = 1.53°. lar velocity, x, The non-coincidence of the axes of inertia and rotation is the reason for the Chandler wobble. Calame (1977) was the first to detect the two larger free libration modes from Lunar laser ranging. Based on 35year series of LLR-measurements, the amplitudes and phases of the Lunar free libration were detected with good accuracy (Williams et al., 2001). According to these data, the amplitudes D1 = 3.3100 , D2 = 8.1900 (Newhall and
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z (C)
y (B)
x (A) Fig. 3. The motion of the inertia axis C relative to the rotation axis due to the Chandler wobble. The ellipse is of wave form due to the FCN.
Williams, 1997), and the period is 74.63 years. This value is close to the one theoretically predicted. This fact once again confirms that the Moon may be well described by the rigid body model with very small elasticity. For the Earth, the Chandler wobble period (433 days) is significantly larger than the Eulerian period (355 days). The full explanation for this discrepancy involves the fluid nature of the Earth’s core and oceans. As for the Free Core Nutation, its appearance in the polar motion is due to a non-coincidence of the rotation axes of the core and total mantle. This mode lies in the Lunar monthly (‘‘Lunar diurnal”) band, but it is a retrograde mode, thus yielding a period smaller than the one Lunar month duration. It will be difficult to detect the FCN, because it blends with more strong resonant harmonics of the forced libration. Moreover the estimated magnitude of PFCN is large and, consequently, the difference of 1 month in (6) is insignificant, i.e. the contribution of the Lunar core is small. Nevertheless, due to the fact that the FCN is in the monthly band, this mode may be of particular interest because it could influence some monthly tides as well as the forced amplitudes–resonance amplification. Certainly, as a whole, the FCN-problem is similar to the terrestrial one, but because of the small sizes of the Lunar core all its effects are much less significant. The forthcoming Japanese experiment ILOM, in which it is planned to establish an optical telescope on one of the poles of the Moon comprises the compact PZT telescope, which will observe the motion of the stars around the pole in order to measure the physical librations of the Moon. It will be fixed to the celestial pole, and equipped with a CCD camera with 4000 4000 pixels to determine the instantaneous position of several tens of stars, accurate to the order of 1 milli-arc-second (1 mas). The first stage of the Japanese mission SELENE (Kaguya) was launched in September, 2007. At the present time it successfully functions. One of the major results of this experiment will be the improvement of a model of the Lunar gravitational field and, in particular, specification of the harmonics of the low order (Goossens et al., 2008; Namiki et al., 2008). All this will allow for signifi-
cantly improving the accuracy of LPhL-theory and hence will raise chances to detect the more fine effects in the Lunar rotation, such as the FCN. The polar telescope (Fig. 3) will show the motion of the Lunar main inertia axis relative to the stars, i.e. relative to the inertial frame. Moreover, from the Lunar surface, it will be possible to detect FCN by those methods that were successfully used for the definition of the FCN of the Earth. In a sense, we will see an opposite picture of the polar motion in the Lunar body if we compare the Moon with the Earth. The amplitude of the Eulerian motions of the Earth’s pole can exceed 0.500 . It is much more than the amplitude of the forced motion, caused by the luni-solar perturbations. That is why only the Eulerian motion is usually taken into consideration when the astronomical observations are being processed. For the Moon, the amplitude of the pole’s forced librations is big and can exceed 20 . The free libration’s amplitude is essentially smaller and does not exceed 900 . Moreover, the free libration’s periods are tens and hundreds of years. This fact should be taken into consideration when the Lunar Navigational Almanac will be developed for future observations at the Lunar surface. When astrometrical observations are processed, the forced motion of the Lunar pole should be taken in the first order; but it is possible not to consider the free motion during 3–5 years. In spite of these differences, the inclusion of the pole’s motion into processing of astrometrical observations is almost the same for the Moon and for the Earth (Kulikov and Gurevich, 1972). Such a fine effect as the FCN has not been observed from the Earth yet. One have to take into account that this fact does not indicate the absence of these phenomena, but it is an evidence of imperfection of the existing methods of processing the observational data. This imperfection consists of the following. Different interpretations of the results are possible, because only those small components of the Moon’s rotation are determined by calculations whose existence was assumed beforehand. For example, the forced libration harmonics or the Chandler-like wobble. Those components, which have not been modeled a priori, cannot be detected. Their existence may be manifested indirectly through decreasing the observational noise. However, it is necessary to point out that if the FCN-mode is detected from observations, then the existence of a Lunar liquid core will be confirmed directly, because only for a planet containing a liquid core, the FCN may be observed. From the LLR, a significant dissipation of the Lunar rotation is detected (Yoder, 1981; Williams et al., 2001). Dissipation is manifested as a small 0.2600 advance in the Lunar rotation axis relative to Cassini’s position. Dissipation from flexing and fluid core interactions should cause the Lunar free polar motion to damp with geologically short time scales. Thus, the detected finite amplitudes imply recent or active stimulation. It has been suggested by Yoder (1981) that this mode may be stimulated by eddies
N. Petrova et al. / Advances in Space Research 42 (2008) 1398–1404
at the solid-mantle/liquid-core boundary. Both damping and active excitation may be observable in the future, and active excitation should cause temporal irregularities in this mode. The free rotational modes for a fluid core and any core–mantle boundary flattening would give rise to a resonant FCN frequency.
1403
CORE
5. Modeling the free periods of the Moon MANTLE
6. Conclusions In this paper, the polar motion for a rigid mantle–liquid core lunar model has been considered. Two modes in its polar motion can be observed in this case. The Chandlerlike mode is already detected from Laser ranging observations (Newhall and Williams, 1997). It manifests itself as an
74,20
Pcw (year)
74,00
4,5 gm/cm 3
73,80
5 5,5
73,60
6 73,40
7
73,20 200
250
300
350
400
450
500
550
600
Rc (km) 144,60 144,40 144,20
PFCN (year)
The modeling was carried out with an intention to detect a dependence of the free libration periods on various parameters characterizing a Lunar liquid core. The parameters of the core – size, ellipticity, densities, and the state of aggregation – were attributed values obtained from the LLR-analyses (Table 1) (Williams et al., 2001). A model of the Moon composed of a rigid mantle and a liquid core of various density values was considered (Fig. 4). The chemical composition of the core was set to vary from a eutectic composition Fe–FeS (with various contents of sulphur) to a purely iron core. The density of the eutectic composition varied from 4 gm/cm3 (a high content of S) up to 7 gm/cm3 (100% Fe). According to Williams et al. (2001), the density of 5.3 gm/cm3 corresponds to 25% wt S. Constraints on the radius of the core at various densities were derived from the fact that the mass of the core should not exceed 4–5% of the Lunar mass. To that end, the normalized moment of inertia C/MR2 has been also calculated and compared to the observed one. As a result, the range from 200 to 600 km was accepted. The core possesses an axial symmetry and has ellipticity, which can be calculated in two ways – either through the mean ellipticity of the mantle’s equator’s, or as a geometrical ellipticity estimated from the LLR-analyses (Dickey et al., 1994). Calculations were made for two values of the core’s ellipticity: ec1 = (ea + eb)/2 = 5.2 104 (the mean dynamical ellipticity of the mantle’s equator), and ec2 ¼ ac ¼ 4 104 (the value of geometrical ellipticity a estimated from LLR-analyses, Dickey et al., 1994). Here a, c are the core’s maximal and minimal semi-axes. In the approximation of a homogeneous core, the dynamical and geometrical ellipticities are equal. The Chandler-like wobble period is independent of the core’s ellipticity, but for the FCN we have obtained for these two ellipticities quite different values: 144 years for ec1 and 186 years for ec2. The dependence of the free periods on the core’s radius for a variety of densities is presented in Fig. 4.
4.5 gm/cm 3
144,00
5 143,80
5,5 143,60
6 143,40
7
143,20 143,00 200
250
300
350
400
450
500
550
600
Rc (km)
Fig. 4. Two-layer model of the Moon. Dependence of the PCW (top) and PFCN (bottom) on the core’s radius for various values of density.
elliptical path of the principal body axis with amplitudes of 8.1900 and 3.3100 . The major axis of the displacement is parallel to the lunar principal B-moment of inertia. The wobble period is 74.63 years. The second mode, Free Core Nutation, can be considered as an additional retrograde elliptical motion over the Chandler’s ellipse with amplitude much smaller than the detected Chandler-like amplitudes, while the period is close to the Lunar sidereal month. According to the classical determination of the FCN-period (Eq. (6)), its magnitude may be estimated by Eq. (7). The forthcoming Japanese experiment ILOM (2013) offers a greater opportunity to detect this mode due to the expected high-accuracy of the observations, and because it will permit the use of methods developed for detection of the terrestrial FCN. Due to the fact that the FCN is in the monthly band, this mode may be of particular interest because it could influence some monthly tides as well as the forced amplitudes–resonance amplification. An analysis of results of the modelling has allowed us to draw the following conclusions:
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1. The dependence on the core’s radius is very weak for both periods (PCW and PFCN), when the radius is set to vary within the range of 300–600 km, the periods are changed by less than 1%. 2. Both periods depend very weakly on the core’s density. Only for radii greater than 400 km, the difference in density is observed. The difference in the periods between the eutectic composition with the density 5.5 gm/cm3 and the pure iron core (7 gm/cm3) is less than 0.04%. 3. The FCN-period is very sensitive to the core’s ellipticity. This property can be used to impose an additional constraint on the core’s parameters, if the expected observation data allows detecting the FCN-mode in the polar motion. The expected amplitudes of the free core’s libration lie within the range of 1–3 ms of an arc (Yoder, 1981). Acknowledgements The authors express their gratitude to Michael Efroimsky, Pini Gurfil and Frank Lemoine for the fruitful discussion and useful advices, which have essentially improved the content of the paper. The research described in this paper was supported by the Russian–Japanese Grant RFFIJSPS N 07-02-91212 (2007–2009). References Barkin, Yu., Gusev, A., Petrova, N. Study of spin–orbit and inner dynamics of the Moon: Lunar mission applications. Adv. Space Res. 37, 72–79, 2006. Calame, O. Free librations of the Moon from Lunar laser ranging, in: Mulholland, J.D. (Ed.), Scientific Applications of Lunar Laser Ranging. Reidel, Dordrecht/Boston, pp. 53–63, 1977. Dickey, J.O., Bender, P.L., Newhall, X.X., et al. Lunar laser ranging: a continuing legacy of the Apollo program. Science 265, 482, 1994. Getino, J. An interpretation of the core–mantle interaction problem. Geophys. J. Int. 120, 693, 1995. Getino, J., Ferrandiz, J.M. A Hamiltonian approach to dissipative phenomena between the Earth’s mantle and core, and effects on free nutations. Geophys. J. Int. 130, 326–334, 1997. Gonzalez, A.-B., Getino, J. The rotation of a non-rigid, non-symmetrical Earth I: free nutations. Celestial Mech. Dyn. Astron. 68, 139–149, 1997. Goossens, S., Matsumoto, K., Liu, Q., Noda, H., Namiki, N., Iwata, T. On the processing of tracking data from the SELENE satellites for orbit and lunar gravity field determination. Abstracts of the 2-nd
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