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Selenographic control

Selenographic control MICHAEL D. MOUTSOULAS; Lunar Science Institute, Houston, Tex., U.S.A. (Received 7 February 1973) Abstract. Evaluation of selenographic data obtained with use of different observational means require the formulation of rigorous algorithms connecting the systems of coordinates, which the various methods have been referred to. The lunar principal axes of inertia are suggested as most appropriate for reference in lunar mapping and selenographic coordinate catalogues. The connection between the instantaneous axis of luna~ rotation (involved in laser ranging, radar studies, astronom- ical observations from the surface of the Moon and VLBI observations of ALSEPs), the ecliptic system of coordinates (which in reductions of observations was considered as fixed in space), the 'Cassini' mean selenographic coordinates (to which physical libration measures were referred), the lunar principal axes of inertia and the invariable plane of the solar system is discussed. The necessity for a frame of reference of uniquely defined selenographic coordinates has already been stressed. The deviation of the shape of the lunar configuration from a sphere is too irregular to suggest any particular points as poles of an ellipsoid. Out of a dozen of geometrical models of the lunar globe, which have appeared so far in the literature based on hypsometric data, there are no two that agree with each other (cf. Kopal, 1969; pp. 191-201). Measure of the lunar limb has been equally inconclusive. The estimations of various investigators for the inclination of the axis of symmetry of the lunar limb lie anywhere between 20 ° and 40 °, while the topographic differences between the northern and southern polar regions was apparently deceptive for the observers and gave rise to the suggestion of most inconsistent non-symmetrical models. The large scale density inhomogeneities in the lunar mass, discovered from perturba- tion analysis of the lunar Orbiters (Muller and Sjogren, 1968) have made obvious that the geometrical shape irregularities were expected. Such been the case, it is suggested that the lunar principal axes of inertia should be considered as the most appropriate system for selenographic reference; they are rigidly fixed in the lunar globe, and therefore the coordinates assigned to each land- mark of the lunar surface will not have to be readjusted with time. However, a consi- derable amount of data in the existing literature refer to a mean position of the lunar axis of rotation, while radar and laser ranging data, as well as Moon-based astrono- mical observations and differential very long base-line interferometry measurements, are related to the instantaneous position of the rotation axis. In order to establish a system of points of selenographic control applicable to the Earth-bound photographic data, to lunar measures performed from an orbiting spacecraft and to ranging results, * Lunar Science Institute Contribution No. 138. ** Communication presented at the Conference on Lunar Dynamics and Observational Coordinate Systems, held January 15-17, 1973, at the Lunar Science Institute, Houston, Tex., U.S.A. * On leave from the University of Manchester, England. The Moon 8 (1973) 461-468. All Rights Reserved Copyright © 1973 by D. Reidel Publishing Company, Dordrecht-Holland 462 MICHAEL D. MOUTSOULAS we need a clear definition of the various frames to which the original observations have been referred. Thus, with application of the appropriate transformation, combination of information obtained by different means can lead to a most accurate and complete control system of selenographic reference. Earth-based photography performed with equatorial telescopes is governed by the orientation of the Earth's axis of rotation at the time, and therefore is obviously referred to the celestial true equatorial system. In the original micrometric lunar measures a technique similar to meridian telescope observations has been used. The times of passages of lunar features and its illuminated limb were recorded for the calculation of differences in right ascension, A~, while micrometric measurements in declination between those features and the northern or southern lunar limb were used for the measurement of differences in declination, A 6. In heliometric observation, too, the lunar limb has been used as reference for the measurement of selenographic posi- tions, with only the difference that the position of the geometrical center of the Moon was determined from measurements of several points along the limb of the lunar disc, and therefore, the interference of the limb irregularities was statistically reduced. The principle of the heliometric technique is illustrated in Figure 1, where R o represents the distance between the adopted center of the lunar disc M and the limb, x and y the cartesian coordinates on the lunar disc (along the directions of the tangents to the equatorial system of selenographic coordinates) of a crater K referred to the center \Ro Fig. 1. Geometry of the heliometric measurements of selenographic coordinates. SELENOGRAPHIC CONTROL 463 M, r the distance between the crater and a point L at the limb, and 0, (p are the posi- tion angles of the point L, referred to the center M and the crater K, respectively. From the relations x = R o cos 0 - r cos ~o, (1) y = Ro sin 0 - r sin q), (2) we obtain by differentiation dx = cos 0 dR o - R o sin 0 dO, (3) dy = sin 0 dRo - Ro cos 0 dO, (4) if we consider that r and qo are actually measured quantities, and therefore in the reduc- tion can be treated as constants. The Equations (3) and (4) give dRo = cos 0 dx + sin 0 dx. (5) If Re is the apparent semidiameter of the lunar disc, computed from the orbital theory, and dRc its deviation from the real value R, while dR o is the difference between R o and R, and therefore R c + dR c = Ro + dRo, Equation (5) gives Rc - Ro = dx cos 0 + dy sin 0 - dRy. (6) By referring the position of the point Kto more than three limb points we can calcu- late the values of the unknown quantities dx, dy and dR c with the least squares method, and finally get the coordinates of the point K with respect to the actual center of the lunar disc. equator "cliptic (Mean celestial I equator ,True celestial equator n equinox ~_/ 'True equinox Fig. 2. Eulerian angles connecting the lunar and celestial equatorial systems. 464 MICHAEL D. MOUTSOULAS It can be easily proved that the angles j, Nf/IL and Lf/IE (Figure 2) which represent the Eulerian angles between the lunar and celestial equatorial coordinate systems can be expressed in terms of elements of the lunar and terrestrial orbit as follows: cosj = cos e cos I + sin e sin I cos , (7) sin I sin ~2 sin Nf4 L = - (S) sinj sin e cos I - cos e sin I cos Q cos NflL - (9) sinj sin L/~E = sin Li~tK cos (l( - ~2) + cos L~rK sin (l( - ~2), (10) cos L fiE = cos Lf/IK cos (l( - f2) - sin LflK sin (l( - f2), (ll) with sin e sin Q sin L~K = (12) sinj and cos e sin i - sin e cos I cos Q cos L/~K = , (13) sinj where l( is the mean longitude of the Moon, t2 the longitude of the ascending node of the lunar orbit, I the inclination of the lunar equator to the ecliptic, e the obliquity of the ecliptic and j the inclination of the lunar equator to the mean celestial equator. The relation between the mean and true equatorial coordinate systems, governed by the perturbed motion of the Earth's axis of rotation, has been fundamental in astrometric studies, and its precise description can be found in numerous publications. However, the method of star-calibrated lunar photography (Moutsoulas, 1970) make possible the direct orientation of a lunar negative with respect to the stellar back- ground. In that way we can avoid reference to the irregularly wandering terrestrial axis, and refer instead to the ecliptic coordinate system (2, fl), which is connected with the equatorial system (~, 3) by means of the well-known formulae cos ;~ cos/~ = cos ~ cos 6, (1.4) cos fl sin 2 = sin ~ cos 6 cos e + sin 6 sin e, (iS) sin fl = - sin ~ cos 6 sin e + sin 3 cos e. (16) The introduction of the lunar motion as basis for a selenographic coordinate system appears to be the cause of some confusion amongst several investigators. Most com- mon mistake has been the assumption that the axis of lunar rotation and its shortest axis of inertia coinside. The orientation of the lunar principal axes of inertia with respect to a coordinate system fixed in space is calculated by means of Euler's dynamical equations. And although for limited time periods (of the order of a year) the ecliptic system of coor- dinates can be considered at first approximation as fixed in space, as the practice has SELENOGRAPHIC CONTROL 465 been in the past, studies of the motion of the Moon about its center of gravity over long periods of time should be referred to the invariable plane of the solar system. The motion of the ecliptic plane in space can be expressed by means of the elements of the spherical triangle .NO0 o (Figure 3). Co Ecliptic of date Ecliptic of epoch plane Fig. 3. Position of the ecliptic plane in space and its motion relative to the invariable plane. From the relations sin 0 sin (f2 o - ZOo) sin (Aco) = , (17) sin i sin 0 cos (f2 o - So) = sin io cos i - cos io sin i cos (Am) = sin(i o - i) + 2 sini cos i o sin 2 (Aco/2), (18) sin (Aco) cos i = sin {£2 - 0to + p)} cos (f2 o - %) - - cos {f2 - 0% + P)} sin (f2 o - %) cos 0 = sin {f2 - (f2 o + p)} + 2 cos {f2 - (~o + P)} x x sin (0 o - %) sin2(O/2), (19) where % and 0 are the longitude of the ascending node and the inclination respectively of the ecliptic of date on the ecliptic of epoch, referred to the mean equinox Yo, P is the general precession in longitude, ~2o, io and f2, i are the longitude of node of the inva- riable plane and its inclination to the ecliptic of epoch and the ecliptic of date and Aa~ is the angular displacement of the intersection of the ecliptic and the invariable plane, the approximate expressions 466 MICHAEL D. MOUTSOULAS 0 sin (0 o - ~o) Aco = , (20) sin io i = io -0 cos (0 o - TOo) + (Aco/2) 2 sin 1" sin (2io), /2 = ~'2 o + p + 0 sin (/20 - re0) cot i - (0/2) 2 sin 1" sin 2 (f20 - zr0), (21) are obtained, which hold an accuracy of the order of 0"0001 per century. Numerical integration of the planetary motions lead to the values i = 1°38'49 " __ 22" (22) f2 = 107013:3 + 2:1 (23) for the inclination, i, of the invariable plane to the ecliptic and the longitude of its node, O, for the year 1950.0. (Clemence and Brouwer, 1955; Woolard and Clemence, 1966). Moreover, use of the differential formulae leads to the expressions sin (O - re) dO dco = sin i ' (24) di = - cos (/-2 - ~r) dO, (25) dr2 = dp + sin (f2 - lr) cot i dO, (26) where zr stands for the longitude of the ascending node of the moving ecliptic on the fixed ecliptic. The inclination, i, is decreasing by 0': 18 yr-1, while the longitude of the node,/2, is increasing by 0~1 yr- 1 Several investigators refer their measures to the 'Cassini coordinates' on the Moon. That system represents the 'mean' orientation of the Moon, i.e., the orientation of a fictitious body obeying precisely the Cassini's laws of motion. According to those (a) the Moon rotates eastward about its polar axis with constant angular velocity in a period of rotation equal to the time of the revolution about the Earth; (b) the incli- nation of the Moon's equator to the plane of the ecliptic is constant; and (c) the poles of the Moon's axis of rotation, of the ecliptic and of the lunar orbit lie in one great circle in the above order, i.e., the planes of the lunar equator, lunar orbit and ecliptic meet in the same line of nodes, the descending node of the equator being at the ascend- ing node of the orbit. (Figure 4) The orientation of a model-moon following Cassini's laws is demonstrated in Figure 5 where X, Y, Z represent the axes of the ecliptic system of coordinates and X', Y', Z' the lunar principal axes, while the Eulerian angles 0, 7/, (p are given as 0 = I, cp = f2, ~ = 180 ° + l( - f2. The velocity components co x, y,, z, are do . dO dr2 cox, = dt sm ~P sin 0 + dt cos 7/= - d~- sin (k -/2) sin/, d~o dO dO coy, = dt cos 7/sin0 - dt sin ~g = - dt cos(l< -/2) sin/, dtP dO dI dq~ cosO+ - (cos/- 1) + , C°z' = d~- dt dt dt CONTROL 467 =,Pole of ecliptic rth lunar pole North celestial pole Fig. 4. Selenocentric celestial sphere. which indicates that even with Cassini's assumptions of orientation the axis of rotation would precess about the direction of fixed inclination to the ecliptic. In the actual motion (referred to the invariable plane) the velocity components along the fixed in space axes are do . dO ~°x = dt sm q) sin 0 + dt cos q~ =d(/~-e+~- ) d(I+O) ~r sin (f2 + a) sin (i + O) + -- cos (f2 + a) dt dt ' d7 / dO (Dy-- - - dt- cos q~ sin 0 + ~ sin q~ d(l(- £2 + ~ - ~) d(I + O) cos (f2 + a) sin (I + 0) + -- cos(Q + a), dt dt d~ dq0 m z = -- cos 0 + -- dt dt d(k- f2 + z - a) d (f2 + a) cos (I + o) + dt dt The solution of this system will be discussed during the Colloquium session on the physical librations. SELENOGRAPHIC 468 MICHAEL D. MOUTSOULAS Z t J I /n I Fig. 5. Eulerian angles of orientation of a model-Moon following Cassini's laws (referred to the ecliptic). Acknowledgments The research reported in this paper was done while the author was a Visiting Scientist at the Lunar Science Institute, which is operated by the Universities Space Research Association under Contract No. NSR 09-051-001 with the National Aeronautics and Space Administration. The capable assistance of the LSI staff at all stages of the work has been invaluable. References Clemence, G. M. and Brouwer, D. : 1955, Astron. J. 60, 118-125. Habibullin, Sh. T. : 1968, Astron. Zh. 45, 663-674. Kopal, Z. : 1969, The Moon, D. Reidel Publ. Co., Dordrecht. Koziel, K.: 1967, Icarus 7, 1-28. Moutsoulas, M.: 1970, The Moon 1, 173-189. Moutsoulas, M.: 1972, The Moon 5, 302-331. Muller, P. M. and Sjogren, W. L.: 1968, Science 161, 680. Woolard, E. W. and Clemence, G. M. : 1966, Spherical Astronomy, Academic Press, New York. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discover Space Springer Journals

Selenographic control

Moutsoulas, Michael D.
Discover Space , Volume 8 (4) – Oct 1, 1973
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Springer Journals
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Copyright © D. Reidel Publishing Company 1973
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2948-2941
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1573-0794
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10.1007/bf00562070
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Abstract

MICHAEL D. MOUTSOULAS; Lunar Science Institute, Houston, Tex., U.S.A. (Received 7 February 1973) Abstract. Evaluation of selenographic data obtained with use of different observational means require the formulation of rigorous algorithms connecting the systems of coordinates, which the various methods have been referred to. The lunar principal axes of inertia are suggested as most appropriate for reference in lunar mapping and selenographic coordinate catalogues. The connection between the instantaneous axis of luna~ rotation (involved in laser ranging, radar studies, astronom- ical observations from the surface of the Moon and VLBI observations of ALSEPs), the ecliptic system of coordinates (which in reductions of observations was considered as fixed in space), the 'Cassini' mean selenographic coordinates (to which physical libration measures were referred), the lunar principal axes of inertia and the invariable plane of the solar system is discussed. The necessity for a frame of reference of uniquely defined selenographic coordinates has already been stressed. The deviation of the shape of the lunar configuration from a sphere is too irregular to suggest any particular points as poles of an ellipsoid. Out of a dozen of geometrical models of the lunar globe, which have appeared so far in the literature based on hypsometric data, there are no two that agree with each other (cf. Kopal, 1969; pp. 191-201). Measure of the lunar limb has been equally inconclusive. The estimations of various investigators for the inclination of the axis of symmetry of the lunar limb lie anywhere between 20 ° and 40 °, while the topographic differences between the northern and southern polar regions was apparently deceptive for the observers and gave rise to the suggestion of most inconsistent non-symmetrical models. The large scale density inhomogeneities in the lunar mass, discovered from perturba- tion analysis of the lunar Orbiters (Muller and Sjogren, 1968) have made obvious that the geometrical shape irregularities were expected. Such been the case, it is suggested that the lunar principal axes of inertia should be considered as the most appropriate system for selenographic reference; they are rigidly fixed in the lunar globe, and therefore the coordinates assigned to each land- mark of the lunar surface will not have to be readjusted with time. However, a consi- derable amount of data in the existing literature refer to a mean position of the lunar axis of rotation, while radar and laser ranging data, as well as Moon-based astrono- mical observations and differential very long base-line interferometry measurements, are related to the instantaneous position of the rotation axis. In order to establish a system of points of selenographic control applicable to the Earth-bound photographic data, to lunar measures performed from an orbiting spacecraft and to ranging results, * Lunar Science Institute Contribution No. 138. ** Communication presented at the Conference on Lunar Dynamics and Observational Coordinate Systems, held January 15-17, 1973, at the Lunar Science Institute, Houston, Tex., U.S.A. * On leave from the University of Manchester, England. The Moon 8 (1973) 461-468. All Rights Reserved Copyright © 1973 by D. Reidel Publishing Company, Dordrecht-Holland 462 MICHAEL D. MOUTSOULAS we need a clear definition of the various frames to which the original observations have been referred. Thus, with application of the appropriate transformation, combination of information obtained by different means can lead to a most accurate and complete control system of selenographic reference. Earth-based photography performed with equatorial telescopes is governed by the orientation of the Earth's axis of rotation at the time, and therefore is obviously referred to the celestial true equatorial system. In the original micrometric lunar measures a technique similar to meridian telescope observations has been used. The times of passages of lunar features and its illuminated limb were recorded for the calculation of differences in right ascension, A~, while micrometric measurements in declination between those features and the northern or southern lunar limb were used for the measurement of differences in declination, A 6. In heliometric observation, too, the lunar limb has been used as reference for the measurement of selenographic posi- tions, with only the difference that the position of the geometrical center of the Moon was determined from measurements of several points along the limb of the lunar disc, and therefore, the interference of the limb irregularities was statistically reduced. The principle of the heliometric technique is illustrated in Figure 1, where R o represents the distance between the adopted center of the lunar disc M and the limb, x and y the cartesian coordinates on the lunar disc (along the directions of the tangents to the equatorial system of selenographic coordinates) of a crater K referred to the center \Ro Fig. 1. Geometry of the heliometric measurements of selenographic coordinates. SELENOGRAPHIC CONTROL 463 M, r the distance between the crater and a point L at the limb, and 0, (p are the posi- tion angles of the point L, referred to the center M and the crater K, respectively. From the relations x = R o cos 0 - r cos ~o, (1) y = Ro sin 0 - r sin q), (2) we obtain by differentiation dx = cos 0 dR o - R o sin 0 dO, (3) dy = sin 0 dRo - Ro cos 0 dO, (4) if we consider that r and qo are actually measured quantities, and therefore in the reduc- tion can be treated as constants. The Equations (3) and (4) give dRo = cos 0 dx + sin 0 dx. (5) If Re is the apparent semidiameter of the lunar disc, computed from the orbital theory, and dRc its deviation from the real value R, while dR o is the difference between R o and R, and therefore R c + dR c = Ro + dRo, Equation (5) gives Rc - Ro = dx cos 0 + dy sin 0 - dRy. (6) By referring the position of the point Kto more than three limb points we can calcu- late the values of the unknown quantities dx, dy and dR c with the least squares method, and finally get the coordinates of the point K with respect to the actual center of the lunar disc. equator "cliptic (Mean celestial I equator ,True celestial equator n equinox ~_/ 'True equinox Fig. 2. Eulerian angles connecting the lunar and celestial equatorial systems. 464 MICHAEL D. MOUTSOULAS It can be easily proved that the angles j, Nf/IL and Lf/IE (Figure 2) which represent the Eulerian angles between the lunar and celestial equatorial coordinate systems can be expressed in terms of elements of the lunar and terrestrial orbit as follows: cosj = cos e cos I + sin e sin I cos , (7) sin I sin ~2 sin Nf4 L = - (S) sinj sin e cos I - cos e sin I cos Q cos NflL - (9) sinj sin L/~E = sin Li~tK cos (l( - ~2) + cos L~rK sin (l( - ~2), (10) cos L fiE = cos Lf/IK cos (l( - f2) - sin LflK sin (l( - f2), (ll) with sin e sin Q sin L~K = (12) sinj and cos e sin i - sin e cos I cos Q cos L/~K = , (13) sinj where l( is the mean longitude of the Moon, t2 the longitude of the ascending node of the lunar orbit, I the inclination of the lunar equator to the ecliptic, e the obliquity of the ecliptic and j the inclination of the lunar equator to the mean celestial equator. The relation between the mean and true equatorial coordinate systems, governed by the perturbed motion of the Earth's axis of rotation, has been fundamental in astrometric studies, and its precise description can be found in numerous publications. However, the method of star-calibrated lunar photography (Moutsoulas, 1970) make possible the direct orientation of a lunar negative with respect to the stellar back- ground. In that way we can avoid reference to the irregularly wandering terrestrial axis, and refer instead to the ecliptic coordinate system (2, fl), which is connected with the equatorial system (~, 3) by means of the well-known formulae cos ;~ cos/~ = cos ~ cos 6, (1.4) cos fl sin 2 = sin ~ cos 6 cos e + sin 6 sin e, (iS) sin fl = - sin ~ cos 6 sin e + sin 3 cos e. (16) The introduction of the lunar motion as basis for a selenographic coordinate system appears to be the cause of some confusion amongst several investigators. Most com- mon mistake has been the assumption that the axis of lunar rotation and its shortest axis of inertia coinside. The orientation of the lunar principal axes of inertia with respect to a coordinate system fixed in space is calculated by means of Euler's dynamical equations. And although for limited time periods (of the order of a year) the ecliptic system of coor- dinates can be considered at first approximation as fixed in space, as the practice has SELENOGRAPHIC CONTROL 465 been in the past, studies of the motion of the Moon about its center of gravity over long periods of time should be referred to the invariable plane of the solar system. The motion of the ecliptic plane in space can be expressed by means of the elements of the spherical triangle .NO0 o (Figure 3). Co Ecliptic of date Ecliptic of epoch plane Fig. 3. Position of the ecliptic plane in space and its motion relative to the invariable plane. From the relations sin 0 sin (f2 o - ZOo) sin (Aco) = , (17) sin i sin 0 cos (f2 o - So) = sin io cos i - cos io sin i cos (Am) = sin(i o - i) + 2 sini cos i o sin 2 (Aco/2), (18) sin (Aco) cos i = sin {£2 - 0to + p)} cos (f2 o - %) - - cos {f2 - 0% + P)} sin (f2 o - %) cos 0 = sin {f2 - (f2 o + p)} + 2 cos {f2 - (~o + P)} x x sin (0 o - %) sin2(O/2), (19) where % and 0 are the longitude of the ascending node and the inclination respectively of the ecliptic of date on the ecliptic of epoch, referred to the mean equinox Yo, P is the general precession in longitude, ~2o, io and f2, i are the longitude of node of the inva- riable plane and its inclination to the ecliptic of epoch and the ecliptic of date and Aa~ is the angular displacement of the intersection of the ecliptic and the invariable plane, the approximate expressions 466 MICHAEL D. MOUTSOULAS 0 sin (0 o - ~o) Aco = , (20) sin io i = io -0 cos (0 o - TOo) + (Aco/2) 2 sin 1" sin (2io), /2 = ~'2 o + p + 0 sin (/20 - re0) cot i - (0/2) 2 sin 1" sin 2 (f20 - zr0), (21) are obtained, which hold an accuracy of the order of 0"0001 per century. Numerical integration of the planetary motions lead to the values i = 1°38'49 " __ 22" (22) f2 = 107013:3 + 2:1 (23) for the inclination, i, of the invariable plane to the ecliptic and the longitude of its node, O, for the year 1950.0. (Clemence and Brouwer, 1955; Woolard and Clemence, 1966). Moreover, use of the differential formulae leads to the expressions sin (O - re) dO dco = sin i ' (24) di = - cos (/-2 - ~r) dO, (25) dr2 = dp + sin (f2 - lr) cot i dO, (26) where zr stands for the longitude of the ascending node of the moving ecliptic on the fixed ecliptic. The inclination, i, is decreasing by 0': 18 yr-1, while the longitude of the node,/2, is increasing by 0~1 yr- 1 Several investigators refer their measures to the 'Cassini coordinates' on the Moon. That system represents the 'mean' orientation of the Moon, i.e., the orientation of a fictitious body obeying precisely the Cassini's laws of motion. According to those (a) the Moon rotates eastward about its polar axis with constant angular velocity in a period of rotation equal to the time of the revolution about the Earth; (b) the incli- nation of the Moon's equator to the plane of the ecliptic is constant; and (c) the poles of the Moon's axis of rotation, of the ecliptic and of the lunar orbit lie in one great circle in the above order, i.e., the planes of the lunar equator, lunar orbit and ecliptic meet in the same line of nodes, the descending node of the equator being at the ascend- ing node of the orbit. (Figure 4) The orientation of a model-moon following Cassini's laws is demonstrated in Figure 5 where X, Y, Z represent the axes of the ecliptic system of coordinates and X', Y', Z' the lunar principal axes, while the Eulerian angles 0, 7/, (p are given as 0 = I, cp = f2, ~ = 180 ° + l( - f2. The velocity components co x, y,, z, are do . dO dr2 cox, = dt sm ~P sin 0 + dt cos 7/= - d~- sin (k -/2) sin/, d~o dO dO coy, = dt cos 7/sin0 - dt sin ~g = - dt cos(l< -/2) sin/, dtP dO dI dq~ cosO+ - (cos/- 1) + , C°z' = d~- dt dt dt CONTROL 467 =,Pole of ecliptic rth lunar pole North celestial pole Fig. 4. Selenocentric celestial sphere. which indicates that even with Cassini's assumptions of orientation the axis of rotation would precess about the direction of fixed inclination to the ecliptic. In the actual motion (referred to the invariable plane) the velocity components along the fixed in space axes are do . dO ~°x = dt sm q) sin 0 + dt cos q~ =d(/~-e+~- ) d(I+O) ~r sin (f2 + a) sin (i + O) + -- cos (f2 + a) dt dt ' d7 / dO (Dy-- - - dt- cos q~ sin 0 + ~ sin q~ d(l(- £2 + ~ - ~) d(I + O) cos (f2 + a) sin (I + 0) + -- cos(Q + a), dt dt d~ dq0 m z = -- cos 0 + -- dt dt d(k- f2 + z - a) d (f2 + a) cos (I + o) + dt dt The solution of this system will be discussed during the Colloquium session on the physical librations. SELENOGRAPHIC 468 MICHAEL D. MOUTSOULAS Z t J I /n I Fig. 5. Eulerian angles of orientation of a model-Moon following Cassini's laws (referred to the ecliptic). Acknowledgments The research reported in this paper was done while the author was a Visiting Scientist at the Lunar Science Institute, which is operated by the Universities Space Research Association under Contract No. NSR 09-051-001 with the National Aeronautics and Space Administration. The capable assistance of the LSI staff at all stages of the work has been invaluable. References Clemence, G. M. and Brouwer, D. : 1955, Astron. J. 60, 118-125. Habibullin, Sh. T. : 1968, Astron. Zh. 45, 663-674. Kopal, Z. : 1969, The Moon, D. Reidel Publ. Co., Dordrecht. Koziel, K.: 1967, Icarus 7, 1-28. Moutsoulas, M.: 1970, The Moon 1, 173-189. Moutsoulas, M.: 1972, The Moon 5, 302-331. Muller, P. M. and Sjogren, W. L.: 1968, Science 161, 680. Woolard, E. W. and Clemence, G. M. : 1966, Spherical Astronomy, Academic Press, New York.

Journal

Discover Space – Springer Journals

Published: Oct 1, 1973

Keywords: Radar; Solar System; Principal Axis; Astronomical Observation; VLBI Observation

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