EPSC Abstracts Vol. 7 EPSC2012-676 2012 European Planetary Science Congress 2012 EEuropeaPn PlanetarSy Science CCongress c Author(s) 2012

Imitation simulating of the lunar physical libration by observing stars from the lunar surface N. Petrova1,2, T. Abdulmyanov2 and H. Hanada3, 1 Kazan Federal University, 18, Kremljevskaja str., Kazan, 420008, Russia, e-mail: [email protected]; 2 Kazan State Power Engineering University, Russia, 3 National Astronomical Observatory, Japan, e-mail: [email protected]

lytical theory is very convenient to carry out such 1. Introduction analysis. We calculated tracks of stars which will Analysis of simulated stellar tracks observable from appear in the field of view of the ILOM-telescope the lunar surface is a kind of a key to the internal during 1 year of observation (2013-2014) for a num- structure of a celestial body. In this connection, the ber of current dynamical models: LLR-model [3]; lunar experiments aimed at the study of the Lunar GLGM-2 (Clementine tracking data) [4]; LP150Q Physical Libration (LPhL) are of great interest. One (Lunar Prospector mission) [5]; SGM100h (SELENE of the necessary stages of preparation for the upcom- tracking data) [6]. ing experiments, such as space mission SELENE-2 Traces of more than 40 stars were analyzed. During [1] the project ILOM (In-situ Lunar Orientation one “lunar day" equal to 27.3 terrestrial days, a star Measurement) is the theoretical simulation of the moves along a spiral (Fig. 1a). In dependence on the future observations. In this report we present several longitude of the star, these spirals can be untwisted results of the simulation of polar stars observation by or twisted. In the latter case a star can describe a loop the imitation of the work of the polar telescope, in the sky of the Moon during the observation period which is planned to be placed on the Lunar pole in (Fig. 1b). The reason of such unusual astrometry the frame of ILOM-project. phenomenon is combination of the slow rotation of the Moon as compared with the Earth and the fast 2. The system of selenographic coordinates precession motion of the lunar pole (in comparison In the framework of the current study we simulate with precession motion of a terrestrial pole). Due to the observation with an “ideal telescope” [2]: the physical libration the shifts of all tracks will be ob- telescope will be posed exactly at the lunar dynami- served towards direction opposite the Earth (Fig. 1c).

300 y (arc sec) 200 cal pole (the axis of its tube coincides with the prin- y (arc sec) 200 Libration is absent cipal inertia axis C of the Moon) and the axes of the 150 100 Libration is presented 100

CCD-array situated in the lens of the telescope will 0 50 ‐500 ‐400 ‐300 ‐200 ‐100 0 100 200 300 x(arc sec) x (arc sec) ‐100 0 be ideally directed along the other two principal axes towards ‐250 ‐200 ‐150 ‐100 ‐50 0 50 100 Earth of inertia A and B. The motion of stars will be dis- ‐200 ‐50 played relatively to the axes of inertia, which are ‐300 ‐100 ‐400 ‐150 rigidly connected with the lunar body. Reduction of a) c) ‐500 ‐200 r rectangular ecliptical coordinates E of any star to 0,20 ) ec r 0,15 s T rc LP‐SGM the selenographycal coordinates (tS = ,,) zyx (a () 0,10 p sss Δ

0,05 may be done with the lunar libration an- t (day) 0,00 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728 glesτ ρ σ ttt )(),(),( on the basis of equation system, ‐0,05

‐0,10 whose common expression can be written in the fol- b) ‐0,15 lowing form [2]: d) r r Fig. 1. Stellar tracs observe from the lunar pole tS = П()σρτ )(),(),()( Ettt (1) The tracks are sensitive to gravity model of the Here П – is a function formed by production of rota- Moon and are different even for the most accurate tion matrixes, used for the transition from the ecliptic modern gravity field models - LP150Q and coordinate system to selenographic system. SGM100h. The difference in numerical values of the

coordinates x (t) and y (t) for these models is more 3. Sensitivity of the tracks to the parameters of s s than 10 milliseconds of arc at any time throughout the dynamical model of the Moon the simulated period (Fig. 1d). Thus, the coordinates In the current study, greater emphasis is placed on of stars in the simulated experiment are sensitive to qualitative effects of the physical libration. The ana- changes in the interior characteristics of the Moon. r 4.Formulation of the inverse problem (inverse problem) we substituted the coordinates S d Under the inverse problem of LPhL we understand о o obtained within the deformable Moon model into Eq. finding the values of libration angles τ (t), ρ (t), σ (2) considering them as observable data and solved o(t) by using “observed” selenographic coordinates r this equation for unknown libration angles X , the of stars ,, zyx ooo measured during the imitation r sss initial values for them X being taken from the rigid observations. In the inverse problem the angles of r r r Moon model X rigid . The residuals d − XX rigid libration are considered as unknown variables de- shown on Fig. 2 point to well-marked both periodical scribed by the vector r T T variations and constant shift (Δτ is insensitive to the = ()σρτ = ( 321 txtxtxttttX )(),(),()(),(),()( ) . changing of model). Then the system of equation (1) can be rewritten:

0,04 r o 0,05 Residuals (arc sec) I + x ) Residuals (arc sec) Δp ΔIσ F )( xxFX 180 ( xI )×+−×+−+= ()( 3 0,03 ∏z ( 31 ) ∏X 2 ∏Z 0,04 0,02 0,03 r r 0,01 0,02 SE =−× 0 0 (2) 0,01 1 51 101 151 201 251 301 351 401 451 501 551 601 651 701 0 -0,01 F is the distance of the mean longitude of the Moon 1 51 101 151 201 251 301 351 401 451 501 551 601 651 701 -0,01 -0,02 from the mean longitude of its ascending (north- -0,02 -0,03 -0,03 -0,04 t(day) t(day) o -0,04 -0,05 ward-bound) node I. The constant I ∼ '5.321 is d r d mean inclination of lunar pole to ecliptic pole. Fig. 2 Residuals in libration angles Δρ = ρ -ρ , Δσ = I(σ - σr) during 26 sidereal months. Jacobian of the system (2) turned out to be close to zero. The systems with the Jacobian close to zero can We carry out the Fast Fourier Transform on the re- be solved using the gradient method [7]. It provides siduals. Obtained spectra for libration angles are r a good convergence for our type of functions F X )( shown on Fig. 3. At the same tame we calculate Δρ and Δσ in analytical form. Strong harmonic 2F with within the given accuracy. the period of 13.62 days is very useful for the analy- Estimation of influence of an inaccuracy in measur- sis. Weak component with the period of 9.077days ing the coordinates on the accuracy of libration an- corresponds to the (l-2F)-term. It may be also inter- gles allows us to do following conclusions: esting for analysis, although its amplitude is on the 1) If the inaccuracy in the determination of coordi- verge of accuracy. We believe theat the constant nates ε = 1 mas is achieved technologically, then the inaccuracy in the determination of LPhL angles will small shifts in libration angles (-0,0117” in Δρ and -0.2619” in Δσ), may be also decreased by improving be ≤Δ 2ερ and I ≤Δ 2εσ . 2) At the same of k2. time, the value of τ t)( is independent on variation in x, y and, consequently, cannot be determined from Δρ ΔIσ the polar stars. This phenomenon is explained by- geometry of physical libration: longitudinal libra- tions depend on selenographic latitude δ proportion- ally to cosδ, which for the polar zone is close to zero.

5.Manifestations of the deformability of the lunar Fig. 3 Spectrum of residuals of Δρ(t) and ΔIσ(t). The unity body in polar libration of ordinate is arc sec. The expansion of Petrova’s analytical theory [8,9] in References the case of a deformable Moon was made on the ba- [1] Hanada H. et al. (2011), Science China. Vol.54. sis of complements, calculated by Chapront et al. [2] Petrova, N., Hanada, H. PSS,. doi:10.1016/j.pss.2011.10.002 [3] Dickey et al., Science, 265: 482,1994); [10] to the Moons’ libration theory [11] concerning [4] Lemoine et al., JGR, v.102, £ E7, p.16,339-16,359, 1997; tidal effects. We have carried out the comparison of [5] Konopliv, NASA Planetary Data System, 2000); r data (coordinates tS )( of the fictitious pole and li- [6] Matsumoto et al., JGR, 115, E06007, 20 pp., 2010; r [7] Demidovich B., Maron I. (2006). Basis of numerical mathe- bration angles X ) obtained in the framework of matics (Book in Russian), 672p. LPhL theory for the rigid Moon and for the deform- [8] Petrova N. (1996) Earth, Moon and Planets, 73, 1, p. 71. [9] Petrova N., Gusev A. (2008) Rotation, physical libration and able Moon respectively (k2 = 0.02992). At the stage interior of the Moon. (Book in Rusian). of direct problem we calculated coordinates and li- [10] Chapront J. et al. (1999) CMDA, 73, 317. bration angles for both models. At the next stage [11] Moons, M. (1982), CMDA, 26, 131.