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Coordinate systems on the Moon and the physical libration: 

Coordinate systems on the Moon and the physical libration Natalia Petrova Kazan state university, Russia 20 September, 2007, Mitaka

Main topics: 

Main topics Celestial systems of coordinate: determination development and selection Lunar rotation and Cassini’s laws Physical libration of the Moon Reductions of stellar coordinates to different systems Definition of orientation of the Moon in space, i.e. the development of coordinate system connected with the Moon, is necessary for studying dynamics of a planetary system Earth-Moon, for mapping lunar surface, for planning and carrying out of experiments on active studying and development of the Moon by means of space robotic technics, for carrying out of reliable astronavigation in space near the Moon and on its surface, for performance of a strict reduction of observation

Celestial ecliptical and equatorial planes: 

Celestial ecliptical and equatorial planes ICRF - Chapront et al., 1999, Astron.Astrophys., 343, 624-633

Position of the vernal Equinox on the stellar sky: 

Position of the vernal Equinox on the stellar sky

Slide5: 

Topographical coordinate system and diurnal motion of stars

Celestial equatorial system of coordinate: 

Celestial equatorial system of coordinate

Relative motion of the system Sun – Earth - Moon: 

Relative motion of the system Sun – Earth - Moon

Cassini's laws: 

Cassini's laws Rotation of the Moon is coordinated with its movement around of the Earth (resonance 1:1) and described by three Cassini's laws (1693). 1. The Moon rotates with constant angular velocity 2. The poles of lunar equator, ecliptic and a lunar orbit lay in one plane (Cassini's plane). The ascending node of a lunar orbit coincides with the descending node of lunar equator 3. The plane of lunar equator is inclined by the constant angle I=1,57о to a plane of ecliptic.

Construction of coordinate systems on the Moon: 

Construction of coordinate systems on the Moon

Construction of coordinate systems on the Moon: 

Construction of coordinate systems on the Moon

Construction of coordinate systems on the Moon: 

Construction of coordinate systems on the Moon

Construction of coordinate systems on the Moon: 

Construction of coordinate systems on the Moon

Construction of coordinate systems on the Moon: 

Construction of coordinate systems on the Moon Porbit  x x x y y y ecliptic equator orbit

Euler angles for description of lunar rotation: 

Euler angles for description of lunar rotation    In the case of Cassini’s motion we have:  =  - precession angle  = 180o+(L- ) - own rotation  = I = 1o32’- nutation angle ecliptic equator orbit

Precession motion of the north and south lunar poles: 

Precession motion of the north and south lunar poles Visual stellar magnitudes North pole: Draco (Dragon) constellation South pole: Dorado (Gold Fish) constellation

Libration of the Moon and angles of libartion: 

Variations of Euler’s angles:  =   = 180o+(L- )  = I Libration of the Moon and angles of libartion

Dynamical and kinematical Euler equations: 

Dynamical and kinematical Euler equations Inertia tensor Moment of external forces = - Vector of instantaneous angular velocity Analytical solution of libration

Coefficients and arguments of libration serirs: 

Coefficients and arguments of libration serirs

Selenocentic coordinate systems: 

Selenocentic coordinate systems Three fundamental directions and reference planes Instantaneous rotation axis  – true instantaneous equator (green) Mean rotation axis 0 – Cassini’s equator (violet) Main axis of inertia C – dynamical equator (red) Three systems of coordinates True selenocentric coordinates Mean selenocentric coordinates A dynamical coordinate system

True selenocentric coordinates (TSC): 

True selenocentric coordinates (TSC) Fundamental direction – instantaneous axes of rotation, fundamental plane – instantaneous equator (x –axis can be directed to the Earth or to vernal equinox). Coordinates of celestial objects, related to the TSC – visible coordinates. selenographical coordinates of an observer from lunar surface and determination of lunar time will be carried out in TSC For the problems of lunar positional astronomy it would be necessary the Astronomical almanac, which will contain the instantaneous places of selected number of stars.

Mean selenocentric coordinates (MSC): 

Mean selenocentric coordinates (MSC) Fundamental direction – Cassini’s (mean) axes of rotation, fundamental plane – mean Cassini’s equator. (x –axis can be directed to the Earth or to vernal equinox). In carrying out astrometric observations from the Moon the coordinates of observing site will have to be determined in just this system The system can by used as intermediate one for correlation between true and visible coordinates of stars .

Dynamical system of coordinates (DSC): 

Dynamical system of coordinates (DSC) Fundamental directions – main axes of inertia – A,B,C. Axis z – is C-axis, x- A-axis The DSC is rigidly connected with a lunar body. The theory of libration determines the motion of DSC relative the inertial SC Position of lunar objects in the DSC is invariable. The DSC can be used as a selenographical SC. At the present time the DSC is the basis for “The Unified Lunar Control Network” (ULCN 2005)- B. A. Archinal et al., 2007, LPC, 1904 Motion of the telescope in ILOM-project may be connected with the DSC-system and will need a minimal reduction to calculated position.

Proposed dislocation of the polar telescope and polar motion: 

Proposed dislocation of the polar telescope and polar motion

Relative motion of the poles (polar motion): 

Relative motion of the poles (polar motion) Nutation motion of dynamical pole relative the mean pole Dynamical pole revolves about the mean pole at the distance  850 m (the principle term of nutation mode is 98”). In comparison for the Earth the same values are  15 m and 0.5.

Reduction of geoequatorial coordinates of a star to the DSC (calculated position): 

Reduction of geoequatorial coordinates of a star to the DSC (calculated position) Take into account the precession motion and proper motion of stars, annual aberration and parallax – true geoequatorial coordinates Transfer from equatorial to ecliptical coordinates Take into consideration the monthly aberration Transfer to DSC, using the theory of physical libration - ((t), (t),(t)).

Stars in a vicinity of lunar world poles: 

Stars in a vicinity of lunar world poles

Conclusion: 

Conclusion Principles of construction of selenicentric reference systems are considered Cassini’s rotation and physical libration are shown in context of coordinate systems. Three kinds of coordinate systems are entered, also their purpose for various problems of a lunar astrometry is considered For the observing of stars in the field of view of the ILOM-telescope the DSC is more suitable The algorithm of reduction of calculated stellar positions to visible place (and inversely) is stated. The idea of construction a Lunar navigational almanac becomes an essential problem for contentious positional observations from the lunar surface

Thank you for attention!: 

Thank you for attention!

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