CC BY
13
UDC 521
DOI: 10.18698/1812-3368-2019-2-4-16
THE PHYSICAL LIBRATIONS OF THE MOON CAUSED BY ITS TIDAL DEFORMATIONS
M.Yu. Barkin1 [email protected]
P.M. Shkapov1 [email protected]
Hideo Hanada2 [email protected]
1 Bauman Moscow State Technical University, Moscow, Russian Federation
2 National Astronomical Observatory of Japan, Oshu, Japan
Abstract
The Moon, like Earth, is not completely solid, and experiences deformation changes, for example due to the tides, caused by the gravitational pull of the Earth's orbit in a complex and resonant nature of the motion of the Moon. It is shown that these deformations lead to temporary variations of Moon inertia tensor components and consequently to the variations in the movement of the poles of the Moon, as well as to the variations of axial rotation. The indicated variations module is in the order of10-12 mas (millisecond of arc). There variations are important for the development of the high-precision theory of lunar physical libration, suitable for modern projects for the reclamation of the Moon, in particular the Japanese project ILOM, which contemplates installing the telescope on the lunar surface and determining its orientation accuracy of the order of 1-0.1 msd, as well as the Russian lunar program, providing the launch of five automatic stations to the Moon in 2019-2024
Keywords
Physical librations of the Moon, tidal deformations, tidal effects, rotation of the Moon, selenopo-tential
Received 26.03.2018 © Author(s), 2019
The research was carried out with the financial support of the Russian Science Foundation (RSFgrant no. 17-71-10254)
Introduction. The research of the physical librations of the Moon caused by its tidal deformations has been performed. The gravitational field of the Moon is considered in the framework of the classical Newton theory. The study is based on the Liouville equations and the equations of motion of a weakly deformable celestial body in Andoyer variables [1, 2]. In this case, the values of temporal (tidal) variations of the coefficients of the second harmonic of the selenopotential, given in Ref. [3], are used. The solution was found by the perturbation method and is presented in analytical form in Andoyer variables;
expressions for the variations of the angular velocity components were also obtained. The amplitudes and periods of physical librations of the Moon caused by its tidal deformations were estimated.
The solution of the Moon's librations task was obtained using two coordinate systems associated with this celestial body Ct,K^kCk and C^C,. The axes of these coordinate systems correspond to the principal central axes of inertia of the Moon, but to different values of the moments of inertia. The corresponding axes of these two coordinate systems coincide and there are simple relations between them: Ct,K =-C£,, C^K = CC,, CQK = C^, which allow establishing relations between two sets of selenopotential constants, defined in the considered coordinate systems. Here, the parameters with the "K" index correspond to the classical selenographic coordinate system Ct,K C, K, and the parameters without an index, to the principal axes of inertia used in solving the Liouville equations in Andoyer variables. For the principle moments of inertia, these relations have the form AK = A, BK = C and CK = B, and for the constants of the second harmonic the selenopotential
JK =-1 (J2-6C22); C2K2 =1 (/2 + 2C22);
2 4 (1)
cK _ 1 c^ rK _ oc . qK _ c S22 C21; C21 - 2S22; S21 -~S21.
The inverse formulas of the specified transformation (1) take the form
J2 =- — (jK-6CK); C22 = — (jK +2Ck> );
2 1 4 (2)
z"1 _ "> qk . o _ r^K. q _ qk C21 -~2s22; s22 - ^ C21; S21 -_S21.
The relations (1) and (2) are used in interpreting the obtained decision on the librations of the Moon due to its tidal deformations. We'll obtain the solution using the parameters (2) without the "K" index, and then convert the resulting solution to the parameters (1) for the classical (basic) selenographic coordinate system C^k ^K CK •
Formulation of the problem. Due to the rotational deformation of the Moon, the polar moment of inertia of a celestial body increases by
5CK =- k
2 , r05ffl
9 f
and the equatorial moments of inertia decrease by the same magnitudes, but 2 times smaller in comparison with 5C [3, 4]. Here © is the angular velocity of
rotation of the Moon; r0 is the average radius of the Moon; k2 is the Love number; f is the gravitational constant.
To describe the temporal variations of the selenopotential coefficients, let's introduce the dimensionless deformation parameter
D _5Ck _ 2 r0V
Dm --J ~~ k2 '
mr02 9 mf
where m is mass of the Moon. At the same time, the elastic Moon rotates as an absolutely rigid body [1, 2], but with modified inertia moments
A = Ak + 3Dromro2; B = BK + 3Dromro2; C = Ck . (3)
Here, AK, BK, CK are the principle moments of inertia of the Moon, which are determined by observations and can be expressed in terms of the mean values of the selenopotential coefficients. If the dimensionless polar moment of inertia I would be introduced by the formula CK = Imr02, then the expressions for the moments of inertia can be obtained from formulas (1)-(3):
Ak = (l + CK - 2CK ) mro2; Bk = (i + C Ko + 2C | ) mro2; C = Imr0. (4)
The expressions for variations of moments of inertia of the Moon can be written if the classical condition for temporal variations of axial moments of inertia 8AK + 8BK + 8CK = 0 is fulfilled (it corresponds to subsurface mass redistribution) for a changeable celestial body
— = — ÔC22--Ô/2; — = — ÔC22--Ô/2; — = — Ô/2;
C I 3I CI 3I C 3I
5F 2 5E 1 SD 1 _
-= —OS22; — = —0C21; -0S21.
CI CI CI
(5)
Tidal variations of the selenopotential coefficients were obtained on the basis of the solution of the classical tidal deformation problem of a celestial body [3]. These periodic variations, caused by the gravitational influence of the Earth, were presented in a standard form
5/f = + X /2V;K) cos0V; 5Cf2 = C(020.°.0;K) + E CVcos
|v| > 1 |v| > 1
8s22 = s2°°.°.°;K) + z S2v2;k)sin0V; sen = c^0.0*) + s C2];k)sin0V^ ^^^
|v| > 1 |v| > 1
8sH = s2010.0.0;k) + z s2rK)cos 0V,
Ivl >1
where J2V;K), C2V;K), ..., S2V;K) are dimensionless amplitudes of the variation of the second harmonic coefficients of the selenopotential; 0V = v1lM + v2lS + + v3F + v4 D is linear combination of the four well-known arguments of the theory of the orbital motion of the Moon with integer coefficients. These arguments are linear functions of time and correspond to constant frequencies nM, ns, nF and nD [5, 6]. In series (6), the constant components of the tidal variations of the selenopotential coefficients are separated, due to the resonant nature of the movement j(«.o.o.oiK), c202°.°.0;k ), ..., s201000;K\
The numerical values of the amplitudes of variations (6) j2v; K), C2V; K), ..., s2V; K) (up to 10"11 rad) are determined in Ref. [3] (the number of trigonometric terms are shown in parentheses): for coefficients J2 (35); C22 (31); s22 (32); C2i(26); S21 (25). The maximum values of these variations are: for variations
/2 (1.545 -10"8 with a period of 27.555 days); C22 (0.722 -10"8 with a period of 27.555 days); S22 (1.039 -10"8 with a period of 27.555 days); C21(2.221 -10"8 with a period of 27.212 days); S21(0.123 -10"8 with a period of 2190.4 days). In the present work, variations (6) are considered as given functions of time. The goal is to identify dynamic effects in the rotational motion of the Moon caused by these tidal variations (6). The equations of motion in Andoyer canonical variables (and their modifications) are used in the form, obtained in [2], for this purpose.
The equations of motion of the Liouville problem in Andoyer variables.
We'll consider the Moon as a weakly deformable body, experiencing deformation due to its own rotation and under the influence of the gravitational attraction of the Earth. In other words, we will consider the Moon as a free body, but with a changeable form and with a time-varying internal structure.
Let's assume that the particles of the body during its movement either slightly deviate from their original positions, or shift in a predetermined manner in time with a small velocity. It can also be assumed that the body has an internal solid shell, with which some Cartesian coordinate system Ct^C, (body axis) is associated, and an external deformable shell. Let Cxyz be the Cartesian coordinate system with the origin at the center of mass of the Moon, maintaining a constant orientation in space. The Andoyer variables will be introduced, which are related to the vector of the angular momentum of the rotational motion of the Moon G [1, 2]
G, e, p, l, g, h (7)
Here for brevity we will not give a detailed description of these variables, but only note that these variables determine the magnitude and orientation of the vector of the angular momentum of the body (Moon) G in two coordinate systems: 1) in the basic ecliptic coordinate system Cxyz; 2) in the intermediate coordinate system CG1G2G3, associated with the vector G and in the axes associated with the heavenly body
In particular, the Andoyer variables 0, l, g are Euler angles that determine the orientation of the axes of inertia of the body in an intermediate coordinate system CG1G2G3, axes of which retain their orientation in space (for an isolated celestial body).
Let's introduce the Andoyer variables L, G, H, where L is projection of the G vector on the polar axis of the body CC,; H is projection of the G vector on the axis Cz; G is the magnitude of the vector G. It is obvious, that
L = G cos 0; G = |G|; H = G cos p. (8)
The variables (8) and l, g, h are canonical, the equations of rotational motion of the considered weakly deformable body in these variables have a canonical form [1, 2]:
dl _dK_ dL __5K ;
dt 8L dt dl
(g dG. (9)
dt dG dt dg
dh H- K•
dt dH' dt dh ' K = 1G2 [(asin2 l + bcos2 l-dsin2l)sin2 0 +
+ c cos2 0-sin 20 (esin l + f cos l)]. (10)
Here, by the condition of the problem, the coefficients a, b, ..., f are known functions of time and are expressed in terms of axial and centrifugal moments of inertia by simple formulas
BC - D2 AC - E2 AB - F2 a =-; b =-; c =-;
AAA
ED + FC DF + BE j FE + AD
f =--:-; e =----; d =----; (11)
AAA
A = ABC - AD2 - BE2 - CF2 - 2DEF.
The Hamiltonian of (10), (11) can be represented as the sum of the zero-order (K0) and perturbed (K0) Hamiltonians. Zero-order Hamiltonian is defined as
K и =1G
Ao B0
sin2 0+—cos2 6 Co
(12)
or
Ko =-G2
±Vg 2 - l2 )+-L L2
Ao Bo Jy } Co
(13)
where A0, B0, C0 are unperturbed values of axial moments of inertia (for example, corresponding to the undeformed state of the Moon).
The Hamiltonian K0 corresponds to the undisturbed rotation of a solid axisymmetric body (or close in its dynamic structure to axisymmetric). In this case, the undisturbed rotational motion of the considered body is described by simple formulas
L = Lo(0 = 0o); l = nit + lo; g = ngt + go,
(14)
where z0 is the initial value of the corresponding variable z = (L, 0, l, g), and the unperturbed frequencies of the Euler body motion are constant and are determined by the formulas
ni = Lo
Ao
Co
ng = 2 Go
ii ( i i
—+-i —+—
Co 2 V Ao Bo
(15)
In the case of the Moon, we'll take its axial rotation around the polar axis of inertia (the axis of the greatest moment of inertia Cr|) as the undisturbed rotational motion. For this unperturbed motion L0 = 0 and, accordingly, the frequency ni = 0.
First-order perturbations in the rotation of the Moon due to its tidal deformations in Andoyer variables. The influence of perturbing factors on the rotation of a deformable body is characterized by a small parameter which is proportional to the magnitude of the variation of the second harmonic coefficients of the selenopotential, i.e., for the Moon p, -10"8. Therefore, it is advisable to use method of the small parameter to solve the considered problem.
Keeping the terms of the first order of smallness, for the perturbing Hamiltonian (the terms of the first order of smallness p.) on the basis of formulas (10), (11), the expression will be obtained
* =1 G2
2 JCo
sin2 01 1 (65C22 + SJ2) sin21 +1 (-65C22 + SJ2) cos21 + 2ÖS22 sin 2l I -
2
~~^J2 cos2 9 + sin29(5C21 sinl + SS21 cosl)
(16)
The selected perturbing Hamiltonian allows to study the dynamic effects in the rotation of a celestial body, caused by cyclical variations or secular changes of the main second harmonic coefficients of the selenopotential 8/2, 8C22, 8C21, 8S2i and 8S22. Thus, first-order disturbances of variables l, g and L (7), (8) are determined by simple quadratures
Sl = dt; SL = dt; Sg = f^1 dt. (17)
J dL J dl J DG
The partial derivatives in integrands (17) are calculated for the unperturbed values of the variables (14), (15) and are known functions of time. For brevity, let's omit the expressions for the indicated partial derivatives and present the results of integration by formulas (17)
Sl =1 X — Si21)sin0V; — = 2 X — Si22)cos0V; 80 = - —; (18) J v >0 ®v G J ||>0 ®v G
Sg = T Z 1 /V2) - 2CV22)^sin0v. (19)
J v||>0 ®v V 3 )
Considering the differential ratio 8L = — sin 080, we obtain another expression for the variation of the variable 0:
§0 = _ 2 £ JL sV22)cos 0v. (20)
J IIVI>0 ®v
This is a complete solution for first-order perturbations in the indicated variables. There will be additional terms in the projections of the angular velocity due to variations in the geometry of the masses of the Moon. There is no data on the variations of its relative kinetic moment (or its components along the axes of inertia of the Moon P, Q and R) for the Moon yet, and the corresponding perturbations in the rotation of the Moon are not considered in the article.
The variations of auxiliary coefficients of the geopotential were used in formulas (18)-(20). They are related to the axes of coordinates O^C, associated with the Moon and directed along the geocentric radius vector of the center of mass of the Moon (axis O^), along the polar axis of inertia O^ and
tangential to the orbit, and in the opposite direction with respect to the direction of motion OÇ. However, in selenodesy, the basic selenographic coordinate system CK^Ç K is used as the main coordinate system associated with the Moon, the axes of which, in a different order, correspond to the above directions: the axis Ct,K is directed to the Earth (CK = -Cthe axis CÇK is the polar axis of inertia (axis CÇK = Cthe axis Ccoincides with the axis CÇ (C= CÇ) and complements the coordinate system to the right.
Formally, let's rename the task parameters. For the coefficients of the second harmonic of the selenopotential, defined in the working coordinate system O^rçÇ, let's retain the usual notation, i.e., J2, C22, S22, C21, S21, and for the same coefficients defined in the main coordinate system of the Moon C^KÇk, we use the new notation (with the "K" index) JK, C^, S^, C2I, S^. It is the last coefficients obtained from satellite and laser observations. Thus, according to the latest observational data for the "Selenium" project, for the average values of these coefficients (for their constant components) their values were determined with high accuracy [5]. The relations between the selenopotential coefficients determined with respect to two coordinate systems are obtained above (i), (2).
As a result, the formulas for the librations of the Moon in variables are written as follows:
5/ = £ s2];K)cos ev = X — s2l;K)cos ev ; (21)
IIVI>0 ®v IVI>0 T
sg = 37 S — J2v;K)sin ev = E T J2v;K)sm 6v ; (22)
3J |v >0 ®v 3J IVI>0 T
50 = 1 S C2v;K)cos 0v = 1 X TV C2V;K)cos 6v. (23)
J ||v >0 ®v J |v >0 T
Here, qv = v1nM + v2nS + v3nF + v4nD; Tv = 2k / qv . Formulas (21)-(23) are complemented by similar trigonometric representations for projections of angular velocity.
The work does not consider the effect of tidal variations in the selenopotential coefficients on the librations of the Moon. Due to the resonant nature of the rotation of the Moon, these perturbing factors are comparable in magnitude with those studied in this paper. In future work, they will be studied separately. For further research, the theory of physical libration of the Moon with a liquid core is of interest [7-9].
Variations of the projections of the vector of angular velocity on the axis of inertia of the body. Let's consider the general formulas connecting the projections of the angular velocity of rotation a = (p, q, r) and the projection of the angular momentum vector G of rotational motion,
= G sin 9 sin l; G^ = G sin 9 cos l; Gq = G cos 9 (24)
on the main (average or unperturbed) inertia axis of the deformable Moon [1] p = a ( G^-P )-f ( Gq- Q )-e ( Gc-R ); q = -f (Q:-P ) + b (Gn-Q )-d (GQ-R ); (25)
r = -e (G^-P )-d (G^ -Q ) + c (Gq-R ),
where coefficients a, b,..., f are known functions of time (11). In the general case, the equations include the projections of the kinetic moment of the Moon mantle particles relative displacements (due to tidal deformations as a result of the attraction of the Earth) onto the average axes of inertia: P, Q and R. There are currently no data on these characteristics of the variable Moon, therefore we take them to be equal to zero.
Keeping only the terms of the first order of smallness with respect to the time variations of the components of the inertia tensor of the Moon (5) and the coefficients of the selenopotential (6), after the necessary transformations of equalities (25), the approximate formulas are obtained
Sp = G \ cos 9 sin lS9 + sin 9 cos l5l - sin 9 sin l + — sin 9 cos l + — cos 9 |; (26) A t ABC J
G f §b ^^ \
Sq = — I cos 9 cos l59 - sin 9 sin lSl--sin 9 cos l h--sin 9 sin l h--cos 9 I; (27)
B ^ B A C J
G / gc ^^ \
5r = — I - sin 959--cos 9 h--sin 9 sin l h--sin 9 cos l I. (28)
CI C A B )
Here, the factors with variations of the Andoyer variables and variations of the components of the inertia tensor depend on the unperturbed values of the variables 9 = ^/2 and I = 0, therefore formulas (26)-(28) can be written in a simplified form
s G SF^ G SB s G ( SD ^ . o.
Sp = — Sl + —I; Sq =---; Sr -S9 +-I, (29)
A
B J 1 B B C
B
or taking into account relations (5) in the accepted approximation
Sp = — f§l + 26S22]; Sq = - — (6ÔC22-ÔJ2);
A ^ J J 3JB^ (30)
§r = — +1SS21
where variation 8p corresponds to the projection of the angular velocity on the equatorial axis of inertia of the Moon C£, the variation ôr corresponds to the projection of the angular velocity onto the other equatorial axis CÇ, variation 8q corresponds to variation in the axial rotation of the Moon and the duration of this rotation (LOD compared with similar characteristics for the Moon). Selenopotential coefficients differ from conventional (classical) notation. Again let's perform the predominance of the axes of inertia according to the rule given earlier, and introduce the classical notation of the projections of the angular velocity
8pK =-8p; 8qK =ôr; 8rK =Sq, (31)
also in accordance with the redesignation of the axes of inertia of the Moon used above. Then finally
^>Pk =y X C2V;K)sin 0v ;
J Ml > 0
SqK = S S2V;K) cos0v; (32)
J Ml > 0
trK=-z J2V;k) cos 0v. 3J MI > 0
The results of the analysis of the physical librations of the Moon caused by its tidal deformations for projections of angular velocity (32) and for variations of the three Andoyer variables (21)-(23) are given in Tables 1-3. It also shows
the amplitudes j2v; k ), C2V; K ), S2V;K ), the periods Tv and arguments 0V for the tidal variations of the second harmonic coefficients of the selenopotential. Table 1 shows the amplitudes, periods and arguments of the tidal variations of the lunar day duration, determined by analogy with the variations of the Earth day duration
HLOD)Moon =~TF .
np
The amplitudes of the variations in the table 3 determined in milliseconds of time (ms).
Table 1
Forced librations of the Moon due to tidal deformations of the Moon, amplitudes and periods of cyclic variations of the projection of the angular velocity and the Andoyer variable
Vi V2 V3 V4 T c2i>, ю-8 pv, 1"T0-3 ev, 1"10-3
0 0 1 0 27.212 -2.2209 11.6421 -11.6421
1 0 1 0 13.691 -0.3028 1.5873 -3.1549
1 0 -1 0 2190.35 0.0613 -0.3213 -0.0040
1 0 -1 -2 14.666 0.0577 -0.3025 -0.0013
0 0 1 2 9.572 -0.0488 0.2558 -0.2538
2 0 1 0 9.146 -0.0355 0.1861 -0.5537
0 0 1 -2 32.281 0.0329 -0.1725 0.1738
1 0 1 2 7.104 -0.0106 0.0556 -0.1100
1 0 1 -2 188.201 -0.0095 0.0498 -0.0994
2 0 -1 0 27.906 0.0052 -0.0273 0.0266
0 0 3 0 9.071 -0.0037 0.0194 -0.0582
0 1 -1 -2 9.829 0.0032 -0.0168 -0.0154
3 0 1 0 6.867 -0.0032 0.0168 -0.0665
2 0 1 -2 24.036 0.0020 -0.0105 0.0313
2 0 1 2 5.648 -0.0014 0.0073 -0.0218
Table 2
Tidal variations of the axial rotation of the Moon 5q (mas) and variations of the Andoye angular variable g (mas)
1m ls F D Tv J2 , 1"-10-3 , 1"-103
0 0 0 0 0 9.446 33.0111 0.0000
1 0 0 0 27.555 1.5453 5.4004 7.9999
1 0 0 -2 31.812 0.2954 1.0323 1.5417
0 0 0 2 14.765 0.2584 0.9030 -0.0108
0 0 0 2 13.606 0.1933 0.6755 -0.0081
2 0 0 0 13.777 0.1267 0.4428 1.3118
1 0 0 2 9.614 0.041 0.1433 0.2105
1 0 2 0 9.108 0.037 0.1293 0.5795
2 0 0 -2 205.892 0.0137 0.0479 0.1424
3 0 0 0 9.185 0.0102 0.0356 0.1584
1 -1 0 0 29.803 0.0095 0.0332 0.0455
0 0 2 -2 173.31 -0.0094 -0.0329 -0.0989
1 0 0 -4 10.085 0.008 0.0280 0.0421
1 1 0 0 25.622 -0.0079 -0.0276 -0.0440
0 0 2 2 7.081 0.0059 0.0206 0.0616
2 0 2 0 6.846 0.0049 0.0171 0.1021
Table 3
Tidal variations of the axial rotation of the Moon 8rK (in milliseconds of arc) and variations of the Andoyer angular variable 8l (in milliseconds of arc)
m lS D Tv SS21 SqK, 1"-10-3 K2 Sl, 1"-10-3
1 0 0 2190.35 -0.1227 0.6432 -0.01242 0.0080
1 0 1 0 13.691 0.1204 -0.6311 1.987576 1.2544
1 0 1 -2 188.201 -0.0283 0.1484 1.995582 -0.2960
2 0 1 0 9.146 0.0205 -0.1075 2.975153 0.3197
0 0 1 2 9.572 0.0188 -0.0986 0.991995 0.0978
0 0 1 -2 32.281 -0.0141 0.0739 1.008005 -0.0745
0 0 1 0 27.212 -0.0105 0.0550 1 -0.0550
2 0 -1 0 27.906 -0.0076 0.0398 0.975153 -0.0388
1 0 1 2 7.104 0.0058 -0.0304 1.979571 0.0602
2 0 1 -2 24.036 -0.0042 0.0220 2.983158 -0.0657
0 0 0 9.071 -0.0037 0.0194 3 -0.0582
3 0 1 0 6.867 0.0032 -0.0168 3.962729 0.0665
0 1 1 0 25.325 -0.0027 0.0142 1.074501 -0.0152
1 -1 1 0 13.197 -0.0016 0.0084 1.913075 -0.0160
2 0 1 2 5.648 0.0014 -0.0073 2.967147 0.0218
1 1 2 1 6.838 -0.001 0.0052 3.058075 -0.0160
Conclusion. Tidal deformations caused by the gravitational influence of the Earth lead to sensitive physical librations of the Moon with amplitudes of the order of a few milliseconds of arc. Variations in the duration of the day due to tidal deformations of the Moon are also significant and can reach tens of milliseconds. The obtained results should be taken into account when implementing lunar projects in the near future, in particular in the Japanese ILOM project for installing a telescope on the lunar surface in order to determine its orientation accurately [10]. High-precision measurements of the orientation of the Moon and its temporal changes open up new possibilities for reseaching in the internal structure and internal dynamics of the Moon, its deformations.
Translated by E. Ovsyannikova
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Barkin M.Yu. — Cand. Sc. (Phys.-Math.), Assoc. Professor, Department of Theoretical Mechanics, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).
Shkapov P.M. — Dr. Sc. (Eng.), Professor, Head of Department of Theoretical Mechanics, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).
Hanada Hideo — PhD, Researcher at the National Observatory of Japan (2-12, Hosh-igaoka, Mizusawa, Oshu Iwate 023-0861, Japan).
Please cite this article as:
Barkin M.Yu., Shkapov P.M., Hanada Hideo. The physical librations of the Moon caused by its tidal deformations. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2019, no. 2, pp. 4-16. DOI: 10.18698/1812-3368-2019-2-4-16
