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Hel2sl(NASA CR OR TMX OR AD NUMBER) (CATEGORI) Microfiche (MF) 4-79 I 1 i On a Refinement of the Theory of the Moon's Physical Librations I. Michelson* * I Abstract i Ihe standard formulation of the dynamics of physical librations of the Moon is re-exsmined in the light of currently accepted reduced estimates af the merhmirnl e1liit.iPit.y af t.he limnr ecuntar- It is neen that a more I I complete mathematical model is required which accounts for centrifugal couples and in which the sum of inclinations of lunar orbit (!jog' ) and equator (1O30') is not regarded as an infinitesimal quantity. Although it remain8 i doubtful whether linearized differential equations can be expected to yield ~ a quantitatively useful theory, a preliminary to more accurate calculation i consists in analyzing the motion with fewer restrictions than has been customary. Ihe main features of such a treatment are given which unifl the classical analysis by showing how the afore-mentioned inclinations can both be used to estimate the two principal mass parameters that affect physical librations. When accurate short-period libration data become available , the constants in question can be evaluated without recourse to orbital data used in the past. GPO PRICE s .~66-20805 8 CFSTI PRICE(S) $ UCCESSION NUMBER) (THRUI -L eL a. t (PAC Hard copy (HC) 2 c w heL2sL(NASA CR OR TMX OR AD NUMBER) (CATEGORI) Microfiche (MF) 4-79 L ff 653 July 65 Professor of Aerospace Engineering, Illinois Institute of Technology, Chicago, Illinois 60616 1 Equations of Physical Libration When the Moon is regarded as a rigid body, its total angular momentum is expressible as a function of its principal-4s inertia moments and angular velocity components: where the principal-exis unit vectors Tir F,, i3 appear with the quantities in each case corresponding to the same axis. The angular velocity components w1, w2, u3 are total quantities, referred to 811 inertial coordinate frame, and it is emphasized that no point of the Moon is then considered to be fixed in space. Taking account of the time variations of the unit vectors, the F1 €2 F3 dw1 a2 dw3 c -=aii A-Fl+B- T2+CKF3 + w1 w2 w3 dt dt dt . (2) Awl Bup CW~ If the components of the total moment of external forces acting are denoted MI, M2, M3, these are equated to the components of (2) to obtain the familiar equations of Euler in scaler form Equations (3) serve as the basis of studies of the physical librations of the Moon, and will be brought to a standard form by evaluating the moments 2 on the right side of (3). For this purpose it is convenient to introduce Cartesian sxes x, y, z, with origin at the Moon's centroid, Ox passing through the Earth's centroid. 'Ihen an element of lunar mass dm situated at (x,y,z) Is at distance p from Earth centroid: p2 = (r - x)2 + y2 + z2 . (4) The Earth-Moon centroid separation is denoted by r, anb the attraction on dmby the Earth mass E considered to act at its centroid is then the units being chosen so that the universal constant of gravitation is unity. With the usual convention for unit vectors r, j, E, the vector location of the mass element is d + yj + 6 and the moment about lunar centroid 0 of the elementary force (5) is The moment is evaluated by integration of (61, using (4) in the form retaining only the first two term in brackets, so that When (8) is referred to principal inertia axes and XI, x2, x3 are understood to represent the Earth's centroid coordinates inthis reference Frame, a straightforward reduction gives 3 where successive terms in (9) give the right sides of the three equations (3). In addition to the approximation already made in neglecting the third and subsequent terms in (7), a further approrimstion is introduced in expressing (9) in terme of ecliptic plane coordinates by considering the angle 8 slibtended by the lunar equator and the ecliptic emall enough to Y ecliptic justim writing Y xp equator cos+ sin+ -esin+ = -sin+ COS+ -8c-4 0 e 1 The transformation (10) shaws that the descending node of the lunar equator on the ecliptic is taben a8 the direction of ads X, while Z is Earth's coordinate normal to the ecliptic plane and XI, x2 are understood to represent the principal axes normal to the Moon's rotation axis. When the orbit inclination i is regarded as a small angle, the coordinate Z is correspondingly limited, and it is customary to neglect the product of the angle 8 with the coordinate Z. In this case it is seen that the products of coordinates (10) required in the mament elrpreesion (9) are of the forms shown on the right sides of the equations (11) 4 .. By replacing the symbols w1, ~2,w3 in (3) by q, r, p respectively, the standard form (11) of the equations of physical librations confornrs with the notation of Laplace (with these two exceptions only, that E and r in present equations are written 88 L and rl in the equation6 identified as (G') in Chapitre 11, Livre Cinquibe, Premikre Partie of the Trait6 de Mecsnique C4leste). The rotatgon of the Moon being principally about the axis, x3, the corresponding component p is larger than either q or r, and Laplace therefore omitted the second term on the left aide in the last one of equations (111, taking the presumed smallness of the principal moment difference (B - A) as further justification for the neglect of this term. For later reference we note here the form of terms neglected in the square brackets of the last equation of (U). When e2 is not neglected, nor the product of 8 and Z, it is readily sham that x3-component of the couple exerted by the Earth on the Moon also contains texms - z2e2 si&+ + o(e3). (12) Libration in Lox@ tude; Estimation of Equatorial Principsl Inertia Moment Difference (B - A) When the product rq in the last of equations (11) is neglected, the equation my be regarded as independent of the two companion equations, and its separate solution is investigated for the purpose of estimating the difference B - A of lunar inertia moments. !he mament term is evaluated first by expressing X and Y in terms of longitude angle v measured from descending node of lunar equator: 5 X = r cosu Y = r sinu (13 It is thus seen that the bracketed term on the right in the last of (11) is given by r2 multiplied with Isin2(u - +)I earth orbit (14) Xf- lunar equator where u - + is the longitude angle s~tendedbythe principal axis xl and the Earth-pointing direction. The Earth's orbital motion relative to the Moon is 6 - $, end if its mean value is denoted by n, while the periodic d inequalities are written as at CH sinn, then Likewise the lunar rotation angle measured from a fixed direction is + - #I, end if this is very nearly equal to the mean orbital motion, we can write +-y=Jnat+u (16) where u is a small quantity taken ES the measure of physical libration in longitude. Thus (15) and (16) show that the Earth's couple is determined by (14) as a term sin2(-u + C H sinn) -2u + 2 C H sinn (17) providedthe terms on the right side of (17) are sufficiently small. men the libration. in longitude is determined by the equation 6 E where is written 88 the motion term n2. Since 7r mean p = + - cosej, 4 - 0, (19) and n may be taken as nearly constant, (16) shows that ana the iioration in longitude is or' fie same character aa kine force4 oscillator: where the inequalities H sin- are regarded as known functions of time. B-A Although the value of the coefficient 3 conthe left side of (21) is not known, the failure to detect librations at frequencies corresponding to the range of values within which it is believed to fall has led to the conclusion that free oscillations are too small to observe. Hence the total libration is attributed to the forcing term and the associated particular solution of the inhomogeneous equation (21): The ellipticity of the Moon's equator being small, and its mechanical measure - , it is seen that appreciable amplitudes can be expected only for great C inequalities (large H) and for long-period inequalities (atdn small). Laplace, having at first no libration measurements with which 7estimates could be inferred, determined that this quantity could be expected to be positive, and less than 0.00284. Later Nicollet's work was taken by Laplace to justim 7 assigning the value 0.000563 in 1819; Frane' corrections reduced this to 0.0003l5 in 1880; Jeffreys in modern times estimates 0.000118 * 0000057, indicating an uncertainty possibly as great as 48% of the estimated value itself. For present purposes it is of main interest to note that the 1819 value, itself a ggsU iraction of the upper bound furnished by Lsplace, rey require a further ten-fold reduction, if current estimates are valid. From B-A (01 1 snrl (001 -e.- +krC ------a --&..ae- -U - --I..-..VQ1-USO LOa- cb%bvAtSprau&~U----..-J-d \GI, UY,--, i+, is oc- YIIP~U d.-WUW-U LSVLULUII UA by like reduction in the values of the libration u. In view of the possibly so appreciable reduction of magnitude of terms retained in (211, the question is raised whether tern earlier neglected in obtaining (11) and deriving (21) from it can still be verified to be uniformly very small by cauparison with tenns retained.
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