ISSN 1063-7729, Astronomy Reports, 2020, Vol. 64, No. 9, pp. 795–803. © Pleiades Publishing, Ltd., 2020. Russian Text © The Author(s), 2020, published in Astronomicheskii Zhurnal, 2020, Vol. 97, No. 9, pp. 765–775. Development of Methods for Navigational Referencing of Circumlunar Spacecrafts to the Selenocentric Dynamic Coordinate System A. O. Andreeva, b, c, Yu. A. Nefedyeva, *, N. Yu. Deminaa, L. A. Nefedieva, N. K. Petrovac, and A. A. Zagidullina a Kazan Federal University, Kazan, 420008 Tatarstan, Russia b Sternberg Astronomical Institute, Moscow State University, Moscow, 119992 Russia c Kazan State Power Engineering University, Kazan, 420066 Tatarstan, Russia *e-mail: [email protected] Received December 19, 2019; revised January 24, 2020; accepted January 24, 2020 Abstract—The initial form of present-day space optical observations contain considerable geometrical and brightness distortions. This problem can be solved based on geometrical correction and transformation of ref- erence object coordinates into commonly accepted cartographic projections. In this work, the method of coordinate transformation to the reference selenocentric dynamic system by using a base of reference sele- nographic objects as electronic maps is considered. The transformation process presupposes the formation of a photogrammetrically corrected image and identification of observed objects included into the electronic maps. Height data of on the Moon’s surface are determined from known observed selenographic coordinates with respect to reference objects by software tools. Preliminary selenographic coordinates can be determined using both ground positional observations and onboard goniometric devices such as a laser interferometer. DOI: 10.1134/S1063772920100017 1. INTRODUCTION for the determination of its position in the phase space In the field of carrying out the system analysis of are related to results of Doppler ground tracking, car- selenophysical data from present-day lunar missions, ried out mainly with the use of the DSN system developing the method for reducing different seleno- (NASA Deep Space Network). Referencing to the physical observations to a common selenocentric sys- DSN system is insufficiently exact, both relative to the tem, and creating a digital catalog of selenocentric ref- celestial coordinate system and to the Earth’s coordi- erence points, one should note the following [1]. At nate systems. Results presented by Konopliv et al. [3] present, in spite of achievements in the determination on laser altimetry of the Moon’s surface and structural of selenophysical parameters based on measurement elements of the Moon’s selenoid are also not refer- data from spacecrafts, the problem of creating exact enced to any specific coordinate system but only ori- coordinate–time orientation on the Moon’s surface ented in a certain way with respect to the Moon’s posi- remains unsolved to a certain degree. According to [2], tion. Each spacecraft revolution around the celestial a conclusion exists that orbital parameters of the body corresponds to a certain orbit. Based on element GRAIL and LRO lunar missions were determined and differences of such orbits at points at which the orbits referenced to the celestial coordinate system with an intersect, the accuracy of coordinate–time referenc- accuracy of 1 m throughout the whole route of the sat- ing for the circumlunar satellite is estimated. In this ellite flight. This conclusion is based on that images case, the determination of an adequate internal accu- received from the LRO board were referenced to each racy of the orbital motion can be discussed. Using other with this accuracy for selenographic coordinates present-day altimetric space measurements and con- when every pixel corresponds to 0.5 m on the Moon’s sidering gravity perturbations improves the matching surface. It is also believed that the gravity field model of orbital parameters and increases the internal accu- constructed by GRAIL measurements is everywhere racy. Still, this does not allow one to estimate the equated and matched with orbital elements of the external accuracy of coordinate–time referencing. LRO mission. However, according to [3] in which Therefore, the described technique can be used only in observations obtained by the LRO spacecraft are ana- the case where the satellite orbits are referenced to lyzed, almost all data on scanning the lunar satellite each other. The LRO mission is also not an exception. 795 796 ANDREEV et al. North Z η M R O +β ξ East Y +λ M ' ζ X Fig. 1. Systems of selenographic coordinates. Estimation of the external accuracy requires support is referencing to a system of light laser beacons another kind of observational data whose accuracy is (LLBs). Such system is similar to the system of refer- not worse than the internal accuracy. Referencing of ence lunar craters; however, the latter are point objects the spacecraft to stars and objects on the Moon’s sur- with which one can implement in principle coordinate face can serve as such observations [4]. In this work, a referencing with an accuracy equal to several centime- method of regression modeling for reducing different ters using the method developed in this work. It is selenophysical measurements to a common seleno- planned to set up such beacons in the course of Rus- centric system and constructing the digital selenocen- sian Luna-25 (26, 27, and 28) lunar missions. Using tric catalog (DSC) of reference lunar objects based on LLBs will make it possible to implement high-accu- their direct referencing to stars and transformation of racy lunar landing and to perform highly detailed data from present-day lunar missions into the DSC scanning of the Moon. In the process of navigational system is proposed. The described results, with allow- referencing of space satellites to the lunar coordinate ance for new theories of physical libration of the system, our method allows one to use both objects Moon, allow one to create a dynamic selenocentric with known coordinates on the Moon’s surface and reference system with coordinate axes that coincide quantum optical systems for which LLBs with deter- with the lunar axes of the Moon’s inertia and the coor- mined exact coordinate positions serve as analogues. dinate origin coincides with the Moon’s center of mass. It should be noted that the first attempt of navi- gational referencing to the digital selenographic cata- 2. SELENOGRAPHIC COORDINATE SYSTEM log in the world practice was made by measurement When creating the reference catalog of lunar systems of the LRO mission. On the platform of this objects of the visible and hidden sides of the Moon spacecraft, the LALT measurement system was (CLO), it is necessary to exactly determine the coordi- mounted. LALT included the LROC main optical nate systems in which all constructions will be per- camera allowing one to obtain lunar surface images formed because there are certain disagreements in the with a resolution of 0.5 m and LOLA altimeter for the special literature on this subject. The rectangular sele- creation of an exact map of height data. The LALT nographic coordinates are reckoned in fractions of the system was used with the aim of determining the most Moon radius R = 1738.1 km. suitable locations for lunar landing of modules of The direction of the axes is as follows (see Fig. 1): future lunar missions. Special attention in implement- ζ is directed to the Earth (x in the Cartesian coordi- ing the LRO project was directed to the increase in the nate system); ξ coincides with the projection of the accuracy of finding the satellite orbit elements. lunar equator and is directed to the East (y in the Car- η High resolution images covering the given lunar tesian coordinate system); and coincides with the territories were obtained using a special panoramic projection of the zero meridian and is directed to the camera. Based on photogrammetric methods and spe- North Pole (z in the Cartesian coordinate system). cial software, an attempt was made to reference the The meridian passing through the first lunar satellite orbit to the digital reference catalog of lunar radius, i.e., through the axis of inertia X , has the lon- objects. A promising direction of coordinate–time gitude of 0°. ASTRONOMY REPORTS Vol. 64 No. 9 2020 DEVELOPMENT OF METHODS FOR NAVIGATIONAL REFERENCING 797 The longitude is reckoned in degrees. According to allows one to increase the accuracy of the transforma- classics of selenography [5, 6], the selenographic lon- tion of selenocentric coordinates more than by 10%. gitude of a point is equal to the dihedral angle between 2. Affine transformation method. This approach is the zero meridian and meridian of the crater. It is pos- used for the transformation of selenocentric coor- itive in the eastern hemisphere and negative in the western hemisphere of the Moon. In both directions, dinates from one Cartesian system X to another sys- longitudes increase to 180° in the first case and tem Y. decrease to –180° in the second case. The meridian of ° The efficiency of the procedures used in the trans- 180 passes through the center of the hidden side of the formation of selenocentric coordinates is determined Moon and traverses Mare Ingenii. However, accord- using the exact comparative analysis. Such procedures ing to [7], the definition is presented incorrectly— include. positive longitudes are reckoned to the West from the zero meridian and negative ones to the East. There- 1. Optimization of polynomial approximations. fore, one must be attentive to publications and defini- 2. Affine transformations. tions which are presented in different sources because a large number of scientific works were published later 3. Orthogonal transformations with enumeration based on them. of systematic errors. The latitude is reckoned in degrees. The arc reck- 4. Solution of linear equation systems. oned from the equator along the meridian passing Different observations of coordinate positions were through the crater to crater is called the latitude.