Coordinate systems on the Moon and Coordinate systems on the Moon and the physical librationthe physical libration

Natalia PetrovaKazan state university, Russia

20 September, 2007, Mitaka

Main topicsMain 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 planesCelestial ecliptical and equatorial planes

ICRF - Chapront et al., 1999, Astron.Astrophys., 343, 624-633

Position of the vernal Equinox on the stellar skyPosition of the vernal Equinox on the stellar sky

Topographical coordinate system and diurnal motion of stars

Celestial equatorial system of coordinateCelestial equatorial system of coordinate

Relative motion of the system Sun – Earth - MoonRelative motion of the system Sun – Earth - Moon

Cassini's lawsCassini's lawsRotation 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 Construction of coordinate systems on the MoonMoon

Construction of coordinate systems on the Construction of coordinate systems on the MoonMoon

Construction of coordinate systems on the Construction of coordinate systems on the MoonMoon

Construction of coordinate systems on the Construction of coordinate systems on the MoonMoon

x (A)

y (B)

Pecliptic

(C )

Construction of coordinate systems on the Construction of coordinate systems on the MoonMoon

x (A)

y (B)

Pecliptic

(C )

Porbit

x

x

x

yyy

ecliptic

equator

orbit

x

y

L

Euler angles for description of lunar rotationEuler angles for description of lunar rotation

26000yrψ

2πT

Earth

precessEarth

In the case of Cassini’s motion we have:

= - precession angle

= 180o+(L- ) - own rotation

= I = 1o32’- nutation angle

18,6yrψ

2πT

Moon

precessMoon

ecliptic

equator

orbit

Precession motion Precession motion of the north and south lunar polesof the north and south lunar poles

Visual stellar magnitudes

North pole: Draco (Dragon) constellation

South pole: Dorado (Gold Fish) constellation

Variations of Euler’s angles:

=

= 180o+(L- )

= I

σ(t)

σ(t)τ(t)

ρ(t)

ninclinitioinlibrationρ(t)

longitudeinlibrationτ(t)

nodeainlibrationσ(t)

Libration of the Moon and angles of libartionLibration of the Moon and angles of libartion

Dynamical and kinematical Euler equationsDynamical and kinematical Euler equations

(t)T

(t)TωIω

dtωId

Tyz ,θ,ψ(θM)(ψМω )

C00

0B0

00A

I

Inertia tensorMoment of external forces =

ω - Vector of instantaneous angular velocity

)sin(),,,( tghQkSCN iinmnmi

)cos(),,,( tbaQkSCK iinmnmi

)cos(),,,( tbaQkSCLI iinmnmi

Analytical solution of libration

Coefficients and arguments of libration serirsCoefficients and arguments of libration serirs

10,110,513,606 d9,9206 d

11,811,818,6 yr14,2273 y

24,624,626,878 d16,827,555 d

78,978,927,212 d16,83 yr

101,399,027,555 d63,9

0.35553.690,71 yr

node(I)inclination

Amplitude ('')Period

Amplitude in longitude

('')Period

Parameters of the forced libration (amplitudes > 10'') (Williams, Dickey, 2003)

10,110,513,606 d9,9206 d

11,811,818,6 yr14,2273 y

24,624,626,878 d16,827,555 d

78,978,927,212 d16,83 yr

101,399,027,555 d63,9

0.35553.690,71 yr

node(I)inclination

Amplitude ('')Period

Amplitude in longitude

('')Period

Parameters of the forced libration (amplitudes > 10'') (Williams, Dickey, 2003)

Selenocentic coordinate systemsSelenocentic coordinate systems

Three fundamental directionsThree fundamental directions and and reference planesreference planes

• Instantaneous rotation axis – true instantaneous equator (green)

• Mean rotation axis 0 – Cassini’s equator (violet)

• Main axis of inertia C – dynamical equator (red)

Pecl

0

C

I

I+

Instantaneous equator

Cassini’s equator

Three systems of coordinatesThree 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 Proposed dislocation of the polar telescope and polar motionand 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.

x ()

y Pecl

o

C

I=1,5o

21

21

6,27

2sin"20

8,27

2sin"98

6,27

2sin"20

8,27

2sin"98

atdays

atdays

atdays

atdays

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

1. Take into account the precession motion and proper motion of stars, annual aberration and parallax – true geoequatorial coordinates

2. Transfer from equatorial to ecliptical coordinates

3. Take into consideration the monthly aberration4. Transfer to DSC, using the theory of physical

libration - ((t), (t),(t)).

Stars in a vicinity of lunar world polesStars in a vicinity of lunar world poles

ConclusionConclusion• 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!