ISSN 1063-7729, Astronomy Reports, 2020, Vol. 64, No. 12, pp. 1078–1086. © Pleiades Publishing, Ltd., 2020. Russian Text © The Author(s), 2020, published in Astronomicheskii Zhurnal, 2020, Vol. 97, No. 12, pp. 1042–1050.

Lunar-Based Measurements of the ’s Physical Libration: Methods and Accuracy Estimates N. K. Petrovaa, b, *, Yu. A. Nefedyeva, A. O. Andreeva, b, and A. A. Zagidullina a Kazan Federal University, Kazan, Russia b Kazan State Power Engineering University, Kazan, Russia *e-mail: [email protected] Received December 19, 2019; revised July 7, 2020; accepted July 30, 2020

Abstract—Computer simulation results are reported for planned lunar-based observations to be conducted using an automated zenith telescope that can be placed at any lunar latitude. Benefits of lunar-based obser- vations are shown, in comparison with lunar laser ranging. It is investigated whether, and how effectively, these observations can be used to determine the lunar rotation parameters (physical libration). The accuracy requirements for these observations are analyzed in view of the accuracy requirements for determining the lunar rotation parameters. The necessary number of telescopes, as well as their optimal locations, is assessed.

DOI: 10.1134/S1063772920120094

1. INTRODUCTION Thus, in 2018, NASA announced the LSITP Lunar-based measurements of physical libration (Lunar Surface Instrument and Technology Payloads) imply that measuring instruments are placed onto the project. This project plans to deliver landing modules lunar surface. Observations from the lunar surface are onto the lunar surface with equipment for completing free from atmospheric fluctuations and need not not only scientific but also commercial tasks. The accommodate the ’s orbital and rotational LSITP is scheduled for implementation in the early motion. 2020s. The new-type observations offer yet another Following the success of the SELENE (Kaguya) advantage—they are independent of lunar laser rang- mission in 2008–2010, Japanese researchers consid- ing (LLR) and can provide even higher accuracy. It is ered the possibility of using a low-orbit satellite and important that we consider here these two matters of lunar-surface radio beacons to implement the inverse principle. Firstly, lunar-based observations are inde- VLBI method. However, computer simulation of this pendent of LLR and, hence, make it possible to clarify experiment [1] showed its low efficiency for determin- whether LLR data have systematic errors. Such errors ing the lunar rotation parameters. may arise because However, another project planned by the Japanese (1) LLR is not sensitive to the direction perpendic- space agency (JAXA)—the ILOM (in situ Lunar Ori- ular to the Earth–Moon line; consequently, observa- entation Measurement)—indeed showed good pros- tions are less sensitive to librations around the Earth– pects in this area. Within the ILOM project, it was Moon line. proposed to put a zenith telescope on one of the lunar poles [2]. It was planned to measure, within that proj- (2) It is difficult to perform LLR during new moon ect, the selenographic coordinates of stars with a high and full moon; therefore, observations are synchro- accuracy and use these data to determine the lunar nized with the libration period. rotation parameters. There are many benefits to posi- (3) LLR amplitudes depend on uncertainties in the tioning an automated zenith telescope (AZT) on a positioning of lunar-based reflectors. pole. Secondly, lunar-based observations enable the sep- 1. Here, good conditions exist for placing measure- aration of different parameters and different frequen- ment instruments due to the existence of both perma- cies. Coupled with an increase in the quality of LLR, nently shadowed and permanently illuminated zones. this separation can significantly improve the quality of 2. Here, stars move slowly, and the number of stars observations over the Moon’s rotation. is limited by the pole’s precession ring, beyond which Space agencies and scientific organizations in the Moon’s diurnal rotation does not take them. All many countries are examining the various types of these factors contribute to efficient detection and such measurements. accurate identification of stars.

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3. The error in obtaining the lunar rotation param- nents of the lunar rotation parameters (LRPs) from eters when processing a single measurement is effec- the observations. tively neutralized by statistics on large stellar samples, The task of enabling the analysis of effects attribut- thus reducing the resulting error to almost zero [3]. able to features of the internal structure of the lunar In [3–6], the possibilities of the ILOM project body necessitates an LRP determination accuracy of were investigated by computer simulation. It was no less than 1″. shown that, apart from suitable conditions for tech- We now define some of the fundamental positions nology positioning of the instruments on the poles, for the localization parameters of the lunar-based tele- this project offers good chances for determining the scope. The position of the telescope in a dynamic lunar rotation parameters in terms of latitude and tilt. coordinate system (DCS) formed by the Moon’s prin- Nevertheless, the described experiment has one cru- cipal axes of inertia is defined by the l and cial setback—as shown by calculations, polar observa- T latitude (Fig. 1). tions cannot be used to determine the third parameter bT of the Moon’s rotation, i.e., longitudinal libration, The position of the DCS relative to, in our case, the which carries in itself a lot of useful information about ecliptic coordinate system is defined by the Euler the lunar structure. angles Ψ, Θ, and ϕ, In this regard, it seems interesting to explore the Ψ=Ω+H ⋅σ, possibilities of an experiment in positioning one or Θ=I +ρ, (1) several AZTs at other lunar latitudes. This paper inves- ϕ= + + °+τ+ ⋅σ, tigates the quality aspects of determinations of the lFT 180 H lunar rotation parameters in the case of nonpolar tele- which include the lunar physical libration (LPL) scope positioning and discusses the search for an opti- angles: τ()t , ρ()t , and Itσ() , in longitude, inclination, mal telescope localization on the Moon. and node, respectively. It is the libration angles that are the LRPs. I is the mean inclination of the lunar Π 2. CONSTRUCTING A MATHEMATICAL pole Plun to the ecliptic pole ecl . In (1), it is assumed MODEL OF THE EXPERIMENT that H =−1 . The parameter F is the latitudinal argu- I The lunar exploration history knows no cases of ment, i.e., the angular distance of the Moon’s mean applying AZTs to determine the lunar rotation param- longitude from the ascending node of the lunar orbit eters; therefore, computer simulation techniques pres- Ω τ ρ σ ent one of the most effective ways to study the funda- . The angles ()t , ()t , and It() for a given time mental possibility, as well as efficiency, of using AZTs interval are calculated using the analytical theory of for these purposes. Simulation of planned observa- physical libration [8, 9]. tions serves to: The field of view of the telescope (FVT) (T ) in (1) Determine the necessary number of AZTs and degrees can be varied, if necessary, in the course of the their optimal positioning. simulation. Since the location of the telescope will change in (2) Develop a program of observations and deter- the course of the simulation, it makes no sense to solve mine their duration. the sampling problem for observations of specific stars (3) Substantiate the accuracy requirements for the from stellar catalogs with all the necessary reductions. observations in view of the accuracy requirements for Therefore, instead of choosing stellar coordinates the lunar rotation parameters. from catalogs, we simulate the ecliptic coordinates of Solving the formulated problems depends, firstly, any given number of stars using a random number on correct construction of the transition matrix generator that simulates the ecliptic λ and between the lunar and inertial coordinate systems and, latitudes β of “stars” in a band of width equal to the secondly, on correct determination of the telescope FVТ, along the line of the precessional motion of the coordinates in the lunar system. telescope (Fig. 2).

The technical characteristics of the telescope in the The system of selenographic stellar coordinates (x1 , simulation are the same as in the ILOM experiment y , z ) in the FVT is defined as follows: the z axis aims ° 1 1 [7]: the field of view is 1 ; the accuracy of one mea- to the zenith of the telescope site T and serves as its surement in the field of view of the CCD matrix is no ″ axis; the x axis goes along the meridian on which worse than 10 ; the telescope has an azimuthal T stands; and the y axis forms a right-hand coordinate mounting, with the tube being directed to the zenith of system (Fig. 1b). the observation site. The duration of the observations depends on the quality of the equipment that main- The position of the telescope is specified relative to tains the operation of the measuring instruments. The the DCS by the longitude lT and latitude bT . We con- desirable duration varies from a year to a year and a sider an ideal situation where the coordinates lT and bT half, so as to be able to refine the long-period compo- are known with absolute accuracy. This idealization is

ASTRONOMY REPORTS Vol. 64 No. 12 2020 1080 PETROVA et al.

Z z(C) y1 П (a)ecl Рlun (b)

y1s Precession Plun and T z1 T

β T Пecl X( ) λT b Ψ T T x1s x1 ϕ Ecliptic Θ = I lT

Equator x(А)

Fig. 1. (a) Position of the telescope Т relative to the ecliptic and dynamic coordinate systems. The lines are shown of the lunar pole’s retrograde precession and the corresponding motion of T. (b) Mapping of the FVT and stellar selenographic coordinates in the telescope coordinate system.

X, deg −2.9 −2.7 −2.5 −2.3 −2.1 −1.9 −1.7

−15.4

to

ecl 2019 to П

2018 −15.9 , deg Y

−16.4

Fig. 2. Motion of the telescope in the ecliptic coordinate system, and the stars whose coordinates were generated in such a way that they fall within the FVT during the designated observation period in 2018–2019. quite far from reality because the coordinates of ya=−cos λ⋅ sin ϕ+ b cos Θ⋅ sin λ⋅ cos ϕ 12 2 (2) objects in the DCS have large errors [10]; thus, such a −Θ⋅ϕ, model can only be regarded as a first approximation. c2 sin cos

Based on these assumptions, we can derive the sel- za13=(cos λ⋅ϕ+Θ⋅λ⋅ϕ cos cos sin sin ) enographic coordinates of a star in the telescope coor- −Θ⋅ϕ+Θ⋅λ+Θ,sinsinsinsincos dinate system (TCS) from the equation system bcd33 3 which was obtained using the rotation matrices in =(cos λ⋅ϕ+Θ⋅λ⋅ϕ cos cos sin sin ) xa11 going from the ecliptic coordinate system to the DCS −bcd11sin Θ⋅ϕ−Θ⋅λ−Θ, sin sin sin 1 cos [4] via the Euler angles (1) and then from the DCS to

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y, s y1, s 2000 2000 (a) (b(b))

15001500 15001515000 StarStar motion directiondirection in FVT 1000 10001000000

500 5005000

−2000 −1500500 −10001000 −500500 0 500 10001000 1500150 2000 −20000000 −1515005000 −1000100000 −5050000 0 500500 10001000 150015000 2000 x, s x , s −505000 −5050000 1

−1000 −10100000000

−1500 −15115005000 02370 2 3 7 −2000 −20200000 0 14 18 2 1717 12 155 1616 20

Fig. 3. (а) Tracks of four polar stars observed with a polar telescope (lT =°0 , bT =°90 ) for 33 days. Star 7 appears in its field of view only periodically. (b) In the field of view of a telescope with the coordinates lT =°0 , bT =°89 , the tracks change their shape; some of the stars appear twice within the specified period.

records are taken quite frequently, every 15 min (see the TCS via the angles lT and bT . Here we introduce the following notation: Fig. 4). =⋅β;=⋅β; ab11sincosTT bb sinsin cb=⋅β;cos cos 4. SOLVING THE INVERSE LPL PROBLEM 1 T FOR NONPOLAR TELESCOPES = ⋅ β; = β; = β; db122cosT sin a cos b cos (3) At the stage of the inverse problem, the rectangular =β;=;=sin 0 cos ⋅β; cos cdab223T coordinates of the star, which were calculated in the bb=cos ⋅ sin β; cb = sin ⋅ cos β; direct problem, are considered as “observed” stellar 33TTcoordinates and are used to calculate the “unknown” =⋅β db3 sinT sin , LRPs. In Eq. (2), these “unknowns”—τ()t , ρ()t , and λ=λ −Ψ λ β Itσ()—are part of the relations for the Euler angles (1) 0 , 0 , are the ecliptic coordinates of the star. and can be derived from these equations by the approximation method [5, 6]. In the course of the simulation, it was shown that the system of nonlinear 3. ANALYSIS OF STAR TRACKS DEPENDING equations (2), despite the close-to-zero Jacobian, ON THE LATITUDE OF THE TELESCOPE SITE allows for a convergent iterative solution by the gradi- The resulting Eqs. (2) and (3) can serve to repro- ent method [10]. duce the construction of star tracks, which is the direct The technology developed by the authors and the problem of the simulation. corresponding software for realizing the gradient If the telescope stands close to the pole, polar stars, method was applied to obtain a stable solution with as seen through it, experience a shift equal to the polar high accuracy both for polar and nonpolar telescope distance of the telescope. In the TCS, the visibility of locations. This result means that if we obtain at the polar stars in its FVT of 1° is shown in Fig. 3. stage of the direct problem for a telescope positioned at The tracks are no longer spirals; within one month, any point on the Moon the selenographic coordinates τ,ρ,σccc a star can cover a part of its path, leave the FVT, and x1, y1 using the LRPs calculated from the then reappear. When moving to southern latitudes, the analytical theory [6], then, at the stage of the inverse pattern of the star tracks undergoes no significant problem, we can introduce in Eq. (2) the coordinates changes; the lines become straighter. Due to the x1, y1 as observed coordinates and thus obtain the LRP increase in the radial velocity of rotation of the tele- values τ,ρ,σooo with an error of less than 0.1″: scope, at low latitudes the instrument is able to detect only 1 to 4 appearances of a star in 2 months, even if ||01τ−τ

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2000 2000 (a)(a) (b) 15001500 1500 1000 10001000 5005 5000

−2000000 −150115000 −10001000 −5500 0 500 10001000 1500 −20020000 −1501151500500 −10001000 −500 0 5005050000 10001000 1500 −500500 −5000 −10010000000 −10010000 −15001505500 −15015000 −2000 −2000

1500 1500 (c)() (d)(d) 1000 1000

5000000 500

−2000000 −150150000 −10001001 00 −5005050000 0 5000000 1000 1500 −2000000 −1500 −1000 −5005050000 0 500 1000 1500150 2000 −50050000 −5000

−100100000000 −10010000

−150015055000 −1500 − 2000 −2000

Fig. 4. The star tracks obtained by solving the direct problem, for stars in the field of view of the AZT, the selenographic coordi- nates of which were calculated with a step of 15 min during the first three hours of two lunar days. The points indicate the time of recording of a given star. The lines are the star tracks. (a) Star observations at the equator (latitude 0°). In this case, there are only 10 stars at the equator, with a maximum of 6 recordings over 3 h. (b) At latitude 30°. In this case, 11 stars are recorded, and there are 8 new appearances. (c) At latitude 60°. In this case, 14 stars are observed, each being recorded a maximum of 10 times. Of these, one star was recorded only once. (d) At the pole (latitude 90°). In this case, due to the slow motion of the stars, all the points recorded over the three hours of observations merged in fact into one big spot. Here, all the stars that fell within the FVT do not leave its visibility field. and I||01σ−σ

ASTRONOMY REPORTS Vol. 64 No. 12 2020 LUNAR-BASED MEASUREMENTS OF THE MOON’S PHYSICAL LIBRATION 1083 generated coordinates in computer memory. This observation regime, i.e., three hours of measurements would have required huge resources of external mem- at the beginning of each month, which allowed us to ory and, no less importantly, would have led to consid- solve the posed problem, i.e., to identify the sensitivity erable expenditure of CPU time, which poses, at the of the LRPs to the measured selenographic coordi- stage of search modeling, a substantial obstacle to the nates of stars at different latitudes. This regime, or analysis of results. If, for example, we took a step of “schedule,” of observations cannot be realized in coordinate recording at 1 h, it turned out that at low practice—it is only the computer that can make leaps latitudes, we could at best record one appearance of a of 27 days in an instant. However, it is this method of star in 1 h of observation (Fig. 3). However, at north- recording stellar coordinates and solving the corre- ern latitudes, the recording of stars can be carried out sponding inverse problem that allowed us to solve the once every 1–2 days since many stars do not leave the problem within a reasonable time. FVT for as long as several months. By applying this observation technique, we were Therefore, in order to obtain a larger number of able, firstly, to record a large number of measurements measurements per one recording of stars and reduce of stars as they crossed the FVT and, secondly, to con- the CPU time per calculation, we found a simple yet siderably reduce the calculation time. For all latitudes, not fully universal way. This approach relies on the fact we obtained the necessary values of both the coordi- that according to our simulations, most stars at lati- nates and LRPs at the same time points, which is an tudes below 75° are visible through a telescope for no important condition for an adequate analysis of the more than three hours. During these three hours, we results. calculated the coordinates every 15 min. When the Thus, we performed calculations for seven possible allotted time expired, we abruptly changed the current latitudes of the telescope site: from 0° to 90° with an time by the duration of the lunar day, i.e., by 27.3 days, interval of 15°. We showed that the longitude of the and again observed the same stars, which reappeared observation site affects only the time of appearance of in the field of the AZT after such a rapid artificial diur- the star in the FVT but does not affect the quality of nal rotation of the Moon. As a result, we conducted a the observations. Therefore, we confined ourselves to triple series of observations of three hours each, considering the positioning of telescopes at the first recording every 15 min the coordinates of the same meridian, where their selenographic longitude is stars in each series (Fig. 4). always zero. In fact, this regime is due to the described feature From the analysis of the tracks presented in Fig. 4, of observation of stars in the case of an azimuthal we see that, at the southern latitudes, the number of mounting of the telescope. To obtain the desired points of recording is smaller than at the middle and result, we had to simulate a process that can hardly be northern latitudes. At latitude ° (the north pole), we reproduced in reality—seven telescopes positioned at 90 see segments of the spirals of almost all the chosen different latitudes simultaneously record the coordi- stars—during the observation time, they circle around nates of a different number of stars for a long time. In the center of the telescope, moving very slowly away order to maintain simultaneity and capture stars at from it as the lunar pole precesses; over three hours of southern latitudes, we need to take records at least observations, the measurement points differ very little every 15 min and save each measurement in RAM or from one other, merging together to create a “blotchy” in files. In one lunar day for one telescope only, pro- feeling. vided that 20–40 stars appear in its field of view per month, the number of these measurements will be In order to test the sensitivity of the LRPs to about 80000. However, in one month, one star (at changes in the selenographic coordinates of the stars, southern latitudes) will yield 2–3 measurements, ± c we randomly changed by 10 ms the values of xs and which, naturally, is not enough for statistics. There- c fore, we had to continue the “observation” process for ys that were obtained from the direct problem. Then, at least three lunar days. As a result, for one telescope, we introduced the changed data into the solution of the calculation took 80–100 min, depending on the the inverse problem and checked how the three LRPs number of stars; most of the time was spent on work- react to the changes in the “observed” coordinates. As ing with files; and it is not possible to store huge data a result, we obtained for each star the LRPs at all the streams in RAM. times of coordinate recording. It should be noted that we, naturally, had no ready- A series of plots (Fig. 5) that demonstrate certain made simulation algorithm. We were developing it in errors (O–C), i.e., differences between the original c c the course of studying the not always readily compre- LRP value, from which xs and ys were calculated, and hensible behavior of stars; therefore, it made little the one obtained from the changed selenographic sense to spend hours waiting for the result in order to coordinates, underpinned our analysis of the simula- analyze the solution and correct the algorithm. When tion results. Figure 5 allows us to compare how the we figured out why stars at different latitudes behave as errors (O–C) change, depending on the observation shown in Fig. 4, we were able to develop the described latitude for all the three LRPs.

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τo − τc, ms 12 (a) 10 8 6 4 2 0 1 3 5 7 9 11 1313 151 1717 19 21 23 25 27 2929 31 33 35 −2 −4 −6 −8 −10 −12 0° 30° 60° 90°

ρo − ρc, ms (b) 16 14 12 10 8 6 4 2 0 −2 1 3 5 7 9 11 133 1515 1717 199 212 233 2525 2727 2929 311 3333 35 −4 −6 −8 −10 −12 −14 −16 0° 30° 6600° 90°

I[σo − σc], ms 16 (c) 14 12 10 8 6 4 2 0 −2 1 3 5 7 9 111 1313 1515 177 191 2121 232 2525 2727 2929 3131 3333 353 37 −4 −6 −8 −10 −12 −14 −16 0° 30° 60° 90°

Fig. 5. Plots of (O–C) in the LRP determinations from one observation at different latitudes of the AZT site—at different times, the “observed stellar coordinates” xs and ys experienced random variations within ±10 ms. The abscissa axis shows the ordinal number of each star measurement, which occurs 15 min after the previous one for all the three series of measurements.

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Having analyzed the residual differences at differ- 5. CONCLUSIONS ent latitudes, we can draw the following conclusions. The modern level of ongoing and planned studies 1. The maximum sensitivity of ξ in the LRPs to of the Moon shows a high accuracy of observations ε and a variety of observational methods, among which changes of in the selenographic coordinates xs and an important place belongs, due to the specific condi- ρ σ ys is obtained for and from observations at the tions on the Moon, to the study of LPLs. This neces- pole: ξ=2 ε , and for τ at the equator ξ=ε . The lon- sitates the development of new methods for analyzing gitudinal libration cannot at all be determined from large arrays of high-precision data on LRP observa- polar observations. tions and extracting from them the maximum amount of useful information about the Moon. In this context, 2. Conversely, equatorial observations provide not the authors’ experience of conducting a computer only good determinations of the longitude libration τ simulation of observations collected using a telescope but also give a possibility to determine the other two placed on the lunar surface shows that this kind of parameters (the libration angles in the tilt ρ and node experiment opens up new possibilities for revealing σ) although they are less sensitive to changes in the subtle effects in the lunar rotation, which, in turn, will selenographic coordinates than in the northern lati- allow scientists to delve into the intricate structure of tudes. If the aim of the observation is to study the lon- the Moon’s interior. gitude libration τ , then AZTs should preferably be The computer simulation was carried out in two placed at a latitude of ±°3 (to avoid shielding by the stages. The first one was the direct simulation problem, Earth, if Т is placed close to the first meridian). i.e., the calculation of the selenographic coordinates of stars on the basis of the selected dynamic model of 3. At a latitude of 30°–45°, all the three LRPs have the lunar body and the LRPs calculated from the ana- virtually the same error ξ≅ε , which, in our opinion, lytical lunar rotation theory constructed for the model makes these latitudes optimal for positioning a tele- of a solid Moon. The second stage was the inverse sim- scope capable of determining all the three LRPs, ulation problem, i.e., the selenographic coordinates which equally respond to variations in selenographic calculated at the first sage were then used to calculate coordinates. the LRPs and analyze the residual differences in com- parison with the original data. The gradient method 4. Predictably, the errors (O–C), obtained by aver- applied for the inverse problem gives a high degree of aging over all the stars at the same time of observation, accuracy. The same stage included the introduction of virtually shrink to zero at all the latitudes due to the controlled errors in the “observed” coordinates, fol- random nature of the changes introduced by us at any lowed by analysis of the sensitivity of the resulting point in time. This result necessitates such a schedule LRPs to these errors. The results of the inverse prob- of future observations that would ensure that as many lem set the foundation for subsequent stages in the stars as possible fall within the FVT at the same time. research, which include the solving of the inverse LPL If the changes in the observed coordinates are not ran- problem, i.e., using the residual differences to refine dom but functional, namely, associated, e.g., with the characteristics of the Moon’s internal structure. applying a flawed model of the Moon’s rotation, the This paper describes a technique for computer sim- average plot might also develop both systematic shifts ulation of observations collected with a lunar AZT, and periodic variations, which in turn may serve as an which can be placed at any point on the lunar surface. observational groundwork for refining the model A substantiation is given for the criteria for determin- parameters. ing the efficiency of the LRP observations. Visual 5. A relative “drawback” of nonpolar AZT observa- materials are presented, which served as a basis for tions, apart from the lack of suitable conditions for the analyzing the research results. measuring equipment and partial shielding of the stars The analysis of the simulated residual differences by the , is the relatively high velocity of stars in the led to the following conclusion. In order to determine FVT, which may interfere with both the recording and the libration angles in the tilt and node, it will be effi- identification of stars. However, bearing in mind that cient to place the AZT at the lunar pole. However, the Moon rotates at a velocity slower by a factor of 27 determining the libration in longitude will require a than the Earth, we can estimate that the velocity of second telescope, which should preferably be placed rotation of stars from observations obtained at the near the equator at a latitude of ±3°. lunar equator corresponds to that of stars at a latitude One telescope can be enough if placed at a latitude of ~88° on the Earth. It is very likely that a technology of 30°–45°, where all the three LRPs will be available of terrestrial observations with such velocities already for determination and equally sensitive to any varia- exists, and if the schedule of forthcoming observations tions in the measured selenographic coordinates. is planned very carefully, there should not be any However, the nonpolar positioning of the telescope issues with the recording and identification of the will require additional engineering research to ensure observed objects. the operability of the measuring equipment as well as

ASTRONOMY REPORTS Vol. 64 No. 12 2020 1086 PETROVA et al. careful elaboration of the stellar identification tech- 2. H. Noda, K. Heki, and H. Hanada, Adv. Space Res. 42, niques and meticulous scheduling of the observations. 358 (2008). 3. N. Petrova, Y. Nefedyev, and H. Hanada, in Proceed- ings of the AIAA SPACE and Astronautics Forum and Ex- FUNDING position SPACE 2016 (2017), p. 5211. 4. N. Petrova and H. Hanada, Planet. Space Sci. 68, 86 This work was supported in part by the Russian Science (2012). Foundation, project no. 20-12-00105 (within which a 5. N. Petrova, T. Abdulmyanov, and H. Hanada, Adv. method for data analysis was developed and numerical cal- Space Res. 50, 1702 (2012). culations were performed). The work was carried out in 6. N. Petrova and H. Hanada, Solar Syst. Res. 47, 463 accordance with the Russian Government Program of (2013). Competitive Growth of Kazan Federal University. This 7. H. Hanada, H. Araki, S. Tazawa, S. Tsuruta, et al., Sci. work was supported in part by the scholarship of the Presi- China Phys., Mech. Astron. 55, 723 (2012). dent of the Russian Federation for young scientists and 8. N. Petrova, Earth, Moon Planets 73, 71 (1996). postgraduate students, no. SP-3225.2018.3, and the Foun- dation for the Advancement of Theoretical Physics and 9. A. V. Gusev, N. K. Petrova, and H. Hanada, Rotation, Physical Libration and Internal Structure of the Active Mathematics “BASIS.” and Multilayered Moon (Kazan. Univ., Kazan’, 2015) [in Russian]. 10. Y. A. Nefedyev, A. Andreev, N. Petrova, N. Y. Demina, REFERENCES and A. Zagidullin, Astron. Rep. 62, 1016 (2018). 1. N. Petrova, A. Gusev, F. Kikuchi, N. Kavano, and H. Hanada, Izv. GAO 219, 262 (2009). Translated by A. Kobkova

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