aa r X i v : . [ phy s i c s . g e o - ph ] A p r On the elements of the Earth’sellipsoid of inertia
Alina-Daniela VˆILCU
Abstract - By using the data for the known geopotential models by meansof artificial satellite, the central moments of inertia of the Earth are determined.For this purpose, it was used the value H = 0 . ± . · − for dynamicalflattening of the Earth [7]. The results obtained indicate that the pole of inertiais located near the Conventional International Origin (CIO). Also, the orienta-tion of the triaxial ellipsoid of inertia for nine geopotential models considered isgiven. Our results improve the ones obtained by Erzhanov and Kalybaev [3]. Key Words and Phrases:
Geopotential, Earth’s moments of inertia, Earth’srotation, Dynamical flattening, Harmonic coefficients.
Mathematical Subject Classification (2000): 85A04, 70F15.
The first artificial satellite of the Earth, Sputnik-1, was launched on 4 October1957. Studying the trajectory of the following satellites (Sputnik-2 and Sputnik-3), D. King-Hele has determined the zonal coefficient J . The value J = 1 . · − was quite close to that calculated by terrestrial measurements. The USsatellite, Vanguard-1, launched in March 1958, made it possible for the firsttime the assessment of discrepancy between the ellipsoid and geoid. The valueof J was obtained in the same year and the first odd zonal term in 1959 by Y.Kozai. In 1961, W. Kaula produced a complete model of degree 4, involving allthe coefficients C lm and S lm of the associated Lagrange function P . From thismoment, the information about the gravitational field of the Earth has becomemore numerous and accurate.The first data from satellites have been used in the development of geopoten-tial models from the early 1970. The SAO-SE model (Smithsonian Astrophys-ical Observatory - Standard Earth), established in 1966, used in 1972 the firstlaser-ranging measurements to establish satellite distances. The GEM model(Goddard Earth Model) was established by NASA’s GSFC (Goddard SpaceFlight Center) in the United States as a reaction to the classified US militarymodels. The first model, GEM-1, was published in 1972, expanding the poten-tial to degree 12. He then followed the whole series of geopotential models to theGEM-10 (developed up to order 20). Subsequently it was developed the modelEGM, as a result of the collaboration between GSFC-NASA, NIMA (National1magery and Mapping Agency) and OSU (Ohio State University). In 1996 cameEGM96S, of degree and order 70, with data provided solely by satellites, andEGM96, of degree and order 360, adjoining geophysical data [1].In this paper, using the data provided by the SE-2 geopotential models fromSE series, GEM-5 to GEM-10 models from GEM series and the EGM96 model,the Earth’s moments of inertia are calculated. Although at most six digitsare accurate, to compare the results more easily with the values obtained byErzhanov and Kalybaev [3], the calculations are performed with nine digits. Inorder to determine the polar moment C ′ it was used the equation obtained byProf. Ieronim Mihaila [6]. In fact, for all the nine models used, the value of C ′ in Tables 4 and 6 coincides with the value of C ′ obtained by considering H ′ = 12 C ′ [2 C ′ − ( A ′ + B ′ )] = H like in [3], where the dynamical flattening of the Earth H = 12 C [2 C − ( A + B )]is obtained from the constant of precession. We used here for dynamical flat-tening the value H = 0 . ± . · − [7]. Thus, it demonstrates thatthe choice made in [3], namely H ′ = H , is valid until the order of 10 − . Theorientation of the ellipsoid of inertia is described in Table 7. The data fromsatellites on the gravitational potential indicate that the equatorial principalmoments of inertia of the Earth are not equal (see Table 6) and it also showsus that the polar axis does not coincide with the axis of rotation. The pole ofinertia P i remains near the Conventional International Origin. In the theory of the movement of the Earth’s artificial satellites it is chosen asa reference system the geocentric system
Oξηζ , the axis Oζ of the system beinggiven by the position of the Conventional International Origin. The origin planfor longitude, Oξζ , is the plan of the Greenwich meridian.In polar coordinates, the expression of the geopotential is [3] U ( r, ϕ, λ ) = G ∞ X n =0 r n +1 n X m =0 δ m ( n − m )!( n + m )! P ( m ) n ( cosϕ ) ×× Z V ( r ′ ) n ρ ( r ′ , ϕ ′ , λ ′ ) P ( m ) n ( cosϕ ′ ) cosm ( λ − λ ′ ) dv or U ( r, ϕ, λ ) = G ∞ X n =0 Y n ( ϕ, λ ) r n +1 , Y n ( ϕ, λ ) = n X m =0 P ( m ) n ( cosϕ )[ A nm cosmλ + B nm sinmλ ] . (1)Here, λ is longitude and ϕ is the geocentric latitude. The symbol V indicatesthat the integration should be extended to the whole volume of the Earth. Thecoefficients A nm and B nm from (1) are A nm = 2 δ m ( n − m )!( n + m )! Z V ( r ′ ) n P ( m ) n ( cosϕ ′ ) cos ( mλ ′ ) ρ ( r ′ , ϕ ′ , λ ′ ) dv, (2) B nm = 2 δ m ( n − m )!( n + m )! Z V ( r ′ ) n P ( m ) n ( cosϕ ′ ) sin ( mλ ′ ) ρ ( r ′ , ϕ ′ , λ ′ ) dv, where δ m = (cid:26) , m ≥ , m = 0 , while P n and P ( m ) n are respectively the conventional zonal harmonics of n th degree and the associated function of Legendre of n th degree and m th order.We mention that the recommended geopotential form by U.A.I. [8] is U ( r, ϕ, λ ) = GMr [1 − ∞ X n =1 ( a e r ) n J n P n ( cosϕ ) ++ ∞ X n =1 ∞ X m =1 ( a e r ) n P ( m ) n ( cosϕ )( C nm cosmλ + S nm sinmλ )] , where the harmonics coefficients of the geopotential are J n = 1 M a en n − m )!( n + m )! Z V ( r ′ ) n P ( m ) n ( cosϕ ) cos ( mλ ′ ) ρ ( r ′ , ϕ ′ , λ ′ ) dv,C nm = 1 M a en n − m )!( n + m )! Z V ( r ′ ) n P ( m ) n ( cosϕ ′ ) cos ( mλ ′ ) ρ ( r ′ , ϕ ′ , λ ′ ) dv, (3) S nm = 1 M a en n − m )!( n + m )! Z V ( r ′ ) n P ( m ) n ( cosϕ ′ ) sin ( mλ ′ ) ρ ( r ′ , ϕ ′ , λ ′ ) dv, while P ( m ) n ( cosϕ ) = 1 √ h nm s δ n ( n − m )!( n + m )! d m P n ( cosϕ ) d ( cosϕ ) m sin m λ. There are three kinds of Legendrians in use: the conventional Legendrianwhen h nm = δ n ( n − m )!( n + m )! , the normalized Legendrian when h nm = 1 and the fully3ormalized Legendrian when h nm = (2 n + 1) − [8]. We use the conventionalLegendrian.The connection between the harmonics coefficients of the geopotential from(3) and the coefficients (2) is given by the relations J n = − M a en A n ,C nm = 1 M a en A nm , (4) S nm = 1 M a en B nm , where M is mass of the Earth.Often, the geopotential is defined by the next expression: U ( r, ϕ, λ ) = GMr [1 + ∞ X n =1 ( a e r ) n I n P n ( cosϕ ) ++ ∞ X n =1 ∞ X m =1 ( a e r ) n I nm P nm ( cosϕ ) cosm ( λ − λ nm )] , where the relations between the coefficients I m , I nm and the constants λ nm withthe coefficients (3) are (see [3]) I n = − J n ,I nm = p C nm + S nm ,λ nm = 1 m arctan S nm C nm . If instead the P ( m ) n Legendre polynomials we consider the functions P nm ,with P nm ( x ) = s n − m )!(2 n + 1)( n + m )! P ( m ) n ( x ) , then the series (3) becomes (see [8]) U ( r, ϕ, λ ) = GMr [1 − ∞ X n =1 ( a e r ) n J n P n ( cosϕ ) ++ ∞ X n =1 ∞ X m =1 ( a e r ) n P nm ( cosϕ )( A nm cosmλ + B nm sinmλ )] , where C nm = A nm s n − m )!(2 n + 1)( n + m )! , nm = B nm s n − m )!(2 n + 1)( n + m )! . If we note the following terms q n = √ n + 1 ,q nm = s n − m )!(2 n + 1)( n + m )! , then the polynomials P m and P nm may be written as follows P n ( x ) = − q n P n ( x ) ,P nm ( x ) = q nm P ( m ) n ( x )and the series (3) becomes (see [8]) U ( r, ϕ, λ ) = GMr [1 −− ∞ X n =1 ∞ X m =0 ( a e r ) n P nm ( cosϕ )( C nm cosmλ + S nm sinmλ )] , where q n C n = I n ,q nm C nm = C nm ,q nm S nm = S nm . The geopotential models are characterized by some constant values (see Ta-ble 1), called the universal constants of geopotential. They include the equato-rial radius of the Earth ( a e ), the geocentric gravitational constant ( GM ) andthe geometrical flattening of the Earth ( f e ).In the following, it is necessary to know the dynamical flattening of the Earth( H ). We use the value of H calculated from the constant of precession [7] H = 0 . ± . · − . (5)MODEL a e [ m ] f e GM · − [ m · s − ]SE-2 6378155.0 1 / .
255 3.986013GEM-5, GEM-6 6378155 1 / .
255 3.986013GEM-7, GEM-8 6378137.8 1 / . / . . ± . / .
257 3.986004415Table 1 . U niversal constants of geopotential Determination of the moments of inertia ofthe Earth
Using the expressions (2) of the harmonics coefficients of the geopotential, onefinds the relations between the first coefficients and the moments of inertia ofthe Earth A ′ , B ′ , C ′ , D ′ , E ′ , F ′ in the system Oξηζ , namely A = A ′ + B ′ − C ′ ,A = E ′ ,B = D ′ , (6) A = B ′ − A ′ ,B = F ′ . For the moments of inertia of the Earth, the following notations were used: A ′ = Z V ρ ( ξ, η, ζ )( η + ζ ) dv,B ′ = Z V ρ ( ξ, η, ζ )( ξ + ζ ) dv, (7) C ′ = Z V ρ ( ξ, η, ζ )( ξ + η ) dv,D ′ = Z V ρ ( ξ, η, ζ ) ζηdv, E ′ = Z V ρ ( ξ, η, ζ ) ζξdv, F ′ = Z V ρ ( ξ, η, ζ ) ξηdv, where ρ is the density. On the other hand, from (4) and (6) the followingrelations between the coefficients I n , C nm , S nm and the moments of inertia ofthe Earth are obtained: M a e I = C ′ − A ′ + B ′ , M a e C = B ′ − A ′ ,M a e C = E ′ , (8) M a e S = D ′ , M a e S = F ′ . The five relations (8) are insufficient to determine the six moments of inertia (7)of the Earth.The system would be complete if another independent equation isadded.Erzhanov and Kalybaev [3] had the idea to use the following expression forthe sixth equation of the system H ′ = 12 C ′ [2 C ′ − ( A ′ + B ′ )] , H ′ = H without a motivation for this approximation. To avoid it,Prof. I. Mihaila deducted an equation for calculating the polar moment C ′ ,using for this purpose the expression of H [7].With this equation of the polar moment: ax + bx + cx + d = 0 , (9)where the coefficients a , b , c and d are respectively a = 8 H ,b = 8 H a ′ ,c = − H (3 − H ) a ′ + 2 H (3 − H ) ( a ′ − b ′ − D ′ − E ′ − F ′ ) ,d = − − H ) a ′ + (3 − H ) a ′ ( a ′ − b ′ − D ′ − E ′ − F ′ ) ++(3 − H ) (2 D ′ E ′ F ′ + a ′ + b ′ E ′ + a ′ − b ′ D ′ ) ,a ′ = − J M a e ,b ′ = B ′ − A ′ = 4 M a e C , the system of six independent algebraic equations (8) and (9) for the six searchedmoments of inertia A ′ , B ′ , C ′ , D ′ , E ′ , F ′ is obtained.If the values of harmonic coefficients of order n = 2 for the geopotential andthe dynamical flattening of the Earth are known, then the normalized momentsof inertia A ′ , B ′ , C ′ , D ′ , E ′ , F ′ can be determined easily. We note here A ′ = A ′ /M a e , etc. The SE − SE − ′ model, where theharmonics coefficients C = − . · − and S = − . · − werecalculated by Erzhanov and Kalybaev [3].MODEL C · C · S · C · S · SE − SE − ′ -484.16596 -0.001196 -0.003466 2.41290 -1.36410 GEM − GEM − GEM − GEM − GEM − GEM −
10 -484.16544 -0.00104 -0.00243 2.43404 -1.39907
EGM
96 -484.16537 -0.000187 0.001195 2.43914 -1.40017Table 2 . N ormalized harmonics coef f icients of the geopotential ( see [3] , [4])7ODEL A ′ B ′ C ′ SE-2 0.329619974 0.329626204 0.330705717SE-2’ 0.329619974 0.329626204 0.330705717GEM-5 0.329620259 0.329626529 0.330706023GEM-6 0.329620507 0.329625671 0.330705717GEM-7 0.329619038 0.329625314 0.330704801GEM-8 0.329619033 0.329625319 0.330704801GEM-9 0.329619643 0.329625927 0.330705412GEM-10 0.329619643 0.329625927 0.330705412EGM96 0.329619636 0.329625934 0.330705412Tables 3 . a . T he normalized moments of inertia of the Earth MODEL D ′ · E ′ · F ′ · SE-2 0 0 -1.761045528SE-2’ -0.004474587 -0.001544029 -1.761045528GEM-5 -0.011231652 -0.001549193 -1.756010649GEM-6 -0.001549193 -0.001161895 -1.792287593GEM-7 -0.001161895 -0.004002083 -1.800420858GEM-8 -0.000387298 -0.000129099 -1.801324554GEM-9 -0.005241437 -0.000271109 -1.804629500GEM-10 -0.003137117 -0.001342634 -1.806191603EGM96 0.001543100 -0.000241400 -1.807607613Tables 3 . b . T he normalized moments of inertia of the Earth In Tables 3, using the models of geopotential from Table 2, we evaluate thesenormalized moments.Further, if the mass M and the equatorial radius of the Earth are known,then one can determine the central moments of inertia A ′ , B ′ , C ′ , D ′ , E ′ , F ′ .In Table 4.a. and Table 4.b., using the data from Table 1 and the value ofthe gravitational constant given by the IAU (1976) System of AstronomicalConstants, namely G = 6 . · − m kg − s − , these moments are evaluated.Once the values of the moments (7) known, the principal moments of inertia A , B , C can be determined by solving the secular equation (see [2], [9])∆( q ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ′ − q − F ′ − E ′ − F ′ B ′ − q − D ′ − E ′ − D ′ C ′ − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 (10)The roots q , q , q of equation (10) represent the principal moments of inertia A , B , respectively C . 8ODEL A ′ · − [ kg · m ] B ′ · − [ kg · m ] C ′ · − [ kg · m ]SE-2 8.010992630 8.011144042 8.037380227SE-2’ 8.010992630 8.011144042 8.037380227GEM-5 8.010999557 8.011151941 8.037387664GEM-6 8.011005584 8.011131088 8.037380227GEM-7 8.010903161 8.011055690 8.037291025GEM-8 8.010903040 8.011055812 8.037291025GEM-9 8.010931380 8.011084104 8.037319434GEM-10 8.010931380 8.011084104 8.037319434EGM96 8.010920187 8.011073251 8.037308375Tables 4 . a . T he moments of inertia of the Earth MODEL D ′ · − [ kg · m ] E ′ · − [ kg · m ] F ′ · − [ kg · m ]SE-2 0 0 -4.279996317SE-2’ -1.087491183 -3.752566228 -4.279996317GEM-5 -2.729709620 -3.765116622 -4.267759686GEM-6 -0.376511662 -2.823837454 -4.355926167GEM-7 -0.282381394 -9.726470762 -4.375656585GEM-8 -0.094127316 -0.313757105 -4.377852885GEM-9 -1.273855995 -0.658891032 -4.385892467GEM-10 -0.762431051 -3.263079405 -4.389688933EGM96 -0.375027747 -0.586687176 -4.393124297Table 4 . b . T he moments of inertia of the Earth MODEL
A B C
SE-2 0.329619513 0.329626665 0.330705717SE-2’ 0.329619513 0.329626665 0.330705717GEM-5 0.329619801 0.329626987 0.330706023GEM-6 0.329619939 0.329626239 0.330705717GEM-7 0.329618555 0.329625797 0.330704801GEM-8 0.329618549 0.329625803 0.330704801GEM-9 0.329619167 0.329626403 0.330705412GEM-10 0.329619160 0.329626410 0.330705412EGM96 0.329619148 0.329626422 0.330705412Table 5 . T he normalized principal moments of inertia of the Earth
The values of the Earth’s normalized principal moments of inertia A , B , C , where A = AMa e etc., obtained by the solving of the equation (10) with thegeopotential models from Tables 3 are presented in Table 5. In Table 6, usingthe data from Table 5 and Table 1, are evaluated the principal moments ofinertia A , B , C . 9ODEL A · − [ kg · m ] B · − [ kg · m ] C · − [ kg · m ]SE-2 8.010981426 8.011155246 8.037380227SE-2’ 8.010981426 8.011155246 8.037380227GEM-5 8.010988425 8.011163072 8.037387664GEM-6 8.010991779 8.011144893 8.037380227GEM-7 8.010891423 8.011067429 8.037291025GEM-8 8.010891277 8.011067575 8.037291025GEM-9 8.010919812 8.011095672 8.037319434GEM-10 8.010919642 8.011095842 8.037319434EGM96 8.010908325 8.011085109 8.037308375Table 6 . T he principal moments of inertia of the Earth As seen from Tables 3 and 5 or from Tables 4 and 6, it is noticed that C ′ coincides with C . For all the nine geopotential models used, the value of C ′ found here coincides with the value of C ′ obtained by considering H ′ = 12 C ′ [2 C ′ − ( A ′ + B ′ )] = H as in the work [3]. It is thus demonstrated that the choice made by Erzhanovand Kalybaev, namely H ′ = H , is valid until 10 − . Let
Oxyz be the system of the Earth’s principal axes of inertia, whose coordi-nate axes are chosen so that A = Z V ρ ( x, y, z )( y + z ) dv,B = Z V ρ ( x, y, z )( z + x ) dv,C = Z V ρ ( x, y, z )( x + y ) dv, Z V ρ ( x, y, z ) xydv = Z V ρ ( x, y, z ) xzdv = Z V ρ ( x, y, z ) yzdv = 0 . The orientation of the system with respect to
Oξηζ may be given by the Eulerangles. We use the notations from [3] β = d ( Oξ, ON ) , α = d ( ON, Ox ) , γ = d ( Oζ, Oz ) , (11)where ON is the intersection between the plans Oξη and
Oxy , called the lineof nodes. 10et ( p ξ , p η , p ζ ), ( q ξ , q η , q ζ ), ( r ξ , r η , r ζ ) be the direction cosines of the axes Ox , Oy , Oz in respect with Oξηζ . They are the projections of the unit vectors p , q , r of the principal axes in the system Oξηζ . On the other hand, the
Oξηζ system overlaps
Oxyz by three rotations R β , R γ , R α . The direction cosine havethe following expressions: p ξ = cos ( x, ξ ) = cosβcosα − sinβsinαcosγ,p η = cos ( x, η ) = sinβcosα + cosβsinαcosγ,p ζ = cos ( x, ζ ) = sinαsinγ,q ξ = cos ( y, ξ ) = − cosβsinα − sinβcosαcosγ,q η = cos ( y, η ) = − sinβsinα + cosβcosαcosγ, (12) q ζ = cos ( y, ζ ) = cosαsinγ,r ξ = cos ( z, ξ ) = sinβsinγ,r η = cos ( z, η ) = − cosβsinγ,r ζ = cos ( z, ζ ) = cosγ and satisfy the orthogonality conditions p · p = q · q = r · r = 1 , (13) p · q = p · r = q · r = 0 . The direction cosines are given by the relations γ i δ i = γ i δ i = γ i δ i = 1 p δ i + δ i + δ i , (14)where δ i , δ i , δ i , with i=1,2,3, are the cofactors of the elements in row i of thedeterminant ∆ which appears in the secular equation (10), q being successivelyreplaced by the principal moments of inertia A , B and respectively C (see [2],[5]). In the relation (14), we have ( γ , γ , γ ) = ( p ξ , p η , p ζ ), ( γ , γ , γ ) =( q ξ , q η , q ζ ), ( γ , γ , γ ) = ( r ξ , r η , r ζ ).In determining the orientation of the principal axes of inertia, besides theorthogonality conditions (13), it is also required to fulfill the condition that thesystem Oxyz to have the same orientation as the system
Oξηζ , namely (see [2]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ξ p η p ζ q ξ q η q ζ r ξ r η r ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 . (15)Once determined the direction cosines, it can be obtained the three Euler’sangles by the relations (12). Thus, for determining the angle γ it is used thefollowing relation cosγ = r ζ , α , the following relations are used p ζ = sinαsinγ,q ζ = cosαsinγ. For the angle β we have the following formulae sinβsinγ = p η q ζ − q η p ζ ,cosβsinγ = p ξ q ζ − q ξ p ζ . If γ = 0, then the axes Oz and Oζ coincide and the plan Oxy coincides with
Oξη . The problem of the orientation for the system
Oxyz is reduced in thiscase to the problem of the orientation for the plan system
Oxy in relation to
Oξη . We have from (13): p ξ = cosβcosα − sinβsinα = cos ( β + α ) = cosλ,p η = sinβcosα + cosβsinα = sin ( β + α ) = sinλ, (16) q ξ = − cosβsinα − sinβcosα = − sin ( β + α ) = − sinλ,q η = − sinβsinα + cosβcosα = cos ( β + α ) = cosλ, where it was noted by λ the angle between axes Ox and Oξ . The expressions(16) give us the ellipse of inertia orientation in the plan Oξη .The orientation of the ellipsoid of inertia corresponding to geopotential mod-els is given in Table 7.a and Table 7.b. From the model SE − γ = 0 wasdetermined the orientation of Oxy in relation to
Oξη . This orientation is usedas a standard for the other models.In Table 7.b, the longitudes and the geocentric latitudes of the ellipsoid ofinertia axes are also given. For the determination of the coordinates λ A and ϕ A of Ox , we have the following relations p ξ = cosλ A cosϕ A ,p η = sinλ A cosϕ A ,p ζ = sinϕ A . Similarly, for the determination of the coordinates λ B and ϕ B of Oy , we havethe following relations q ξ = cosλ B cosϕ B ,q η = sinλ B cosϕ B ,q ζ = sinϕ B . For Oz , the longitude λ C is determined from the following formula λ C = 360 ◦ − (90 ◦ + β ) , where β is given by (11). 12ODEL α ◦ β ◦ γ (in arcsec (”))SE-2 - - 0SE-2’ 4.3 -19 0.9GEM-5 -7 -7.9 2.2GEM-6 19.4 -36.8 0.4GEM-7 58.7 -73.6 0.8GEM-8 3.5 -18.4 0.1GEM-9 -11.9 -3.0 1.0GEM-10 8.2 -23.1 0.7EGM96 23.7 -38.6 0.3Table 7 . a . T he orientation of the ellipsoid of inertia MODEL λ A ◦ ϕ A ◦ λ B ◦ ϕ B ◦ λ C ◦ ( π − ϕ C )”SE-2 -14.7 0 75.3 0 0 0SE-2’ -14.7 1 . · − . · −
251 0.9GEM-5 -14.9 − . · − . · − . · − . · − . · − . · − . · − . · − − . · − . · − . · − . · − . · − . · − . b . T he orientation of the ellipsoid of inertia As it is observed in Table 7.b, for the geopotential models considered, thelongitude of Ox axis of the triaxial ellipsoid of inertia is about − ◦ and thelongitude of Oy axis has a value close to 75 ◦ , except for the model GEM − λ A ≃ − ◦ and λ B ≃ ◦ . From the values obtained, except for themodel GEM-5, it is determined the mean ellipsoid of inertia with the principalmoments A = 8 . · kg · m , B = 8 . · kg · m , C =8 . · kg · m and with the orientation λ A = − ◦ . λ B = 74 ◦ . H obtained from the mean values of the principal moments ofinertia, namely H = 0 . P for the ninemodels of geopotential considered. Since the angle γ is a small angle, then, in aCartesian reference XP Y in the tangent plan at P , with the axis P X tangentto the Greenwich meridian, the pole of inertia P i will have the polar coordinates( λ C , γ ).It is remarked that the pole of the ellipsoid SE-2 when γ = 0 coincides withthe conventional international pole P , the other positions of the pole remainingin the neighborhood, except for the pole of inertia corresponding to GEM − P are λ C = 209 ◦ . γ = 0” .
5. Therefore13he mean polar axis differs from the rotation axis by 0” .
5. The mean pole P deviates approximately by 15 meters from the pole of rotation. YX , G b bbbbbb bb P Figure 1 . T he position of the inertial pole on the surf ace of the Earth
The results obtained show that the approximation made in the paper [3] issatisfied and the improved values for the principal moments of inertia A , B , C are obtained. On the other hand, it is better emphasized the fact that the polaraxis of inertia is located in the neighborhood of the Earth’s rotation axis. Forthe geopotential models considered, the longitudes of the axes Ox and Oy ofthe triaxial ellipsoids of inertia have concordant values. Acknowledgements . I wish to express my deep gratitude to my advisor,Professor Ieronim Mihaila from the University of Bucharest, who encouragedand assisted me in the development and completion of this paper.
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