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Lunar-Based Measurements of the Moon's Physical Libration: Methods

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Lunar-Based Measurements of the Moon's Physical Libration: Methods 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 Moon’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 Earth’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. 1078 LUNAR-BASED MEASUREMENTS OF THE MOON’S PHYSICAL LIBRATION 1079 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 longitude 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 longitudes 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.
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