- Email: [email protected]
The values of the nutations and of the period of the variation of latitude
T h e V a l u e s of t h e N u t a t i o n s a n d of t h e P e r i o d of t h e V a r i a t i o n of L a t i t u d e R. O. VIC~NTE Faculty of Sciences, Lisbon, Portugal SUMMARY The values of t h e n u t a t i o n s can be determined f r o m t h e observations and also f r o m theoretical expressions deduced in the theory of the rotation of the Earth. It is pointed out that these values are not in
agreement because the theoretical calculations did not take into account the internal constitution of the Earth. A theory of the nutation is described that gives values for the nutations and the period of latitude variation in better agreement with the observations than the previous theories because it considers the elastic properties and internal constitution of the Earth. Some possible improvements in the observational and theoretical aspects of the problem are mentioned.
1. INTI~ODUCTION THE d e t e r m i n a t i o n of t h e values of t h e several n u t a t i o n s is one of those p r o b l e m s in a s t r o n o m y which offer good e x a m p l e s of a d v a n c e s m a d e in successive periods of t h e o r e t i c a l a n d of o b s e r v a t i o n a l researches. T h e o b s e r v a t i o n s led to results which were n o t e x p e c t e d theoretically, a d i s c r e p a n c y caused b y over-simplified h y p o t h e s e s . T h e n u t a t i o n due to t h e S u n ' s influence was first n o t e d b y NEWTO:N ; t h e irregularities caused b y t h e m o t i o n of t h e Moon's nodes were discovered b y BRADLEY f r o m a r e m a r k a b l e series of o b s e r v a t i o n s lasting eighteen years. T h e theoretical e x p l a n a t i o n of t h e n u t a t i o n s was g i v e n b y D'ALEMBERT, s h o r t l y a f t e r B~ADLEY'S results were published. T h e t h e o r y was p e r f e c t e d b y EULER a n d LAPLACE. Since t h e n n u m e r o u s investigations h a v e b e e n m a d e ; t h e latest results, giving full expressions for t h e n u t a t i o n s a n d t h e i r n u m e r i c a l values, were p u b l i s h e d b y E. W. WOOLARD (1953). W h e n , in t h e solution of t h e d y n a m i c a l equations of motion, we do n o t consider t h e d i s t u r b i n g forces, a n d r e g a r d t h e E a r t h as s i m p l y set in r o t a t i o n a n d left to itself, t h e result o b t a i n e d is a small oscillation of t h e E a r t h ' s axis, which is called t h e free E u l e r i a n n u t a t i o n . T h e initial theoretical researches showed t h a t this n u t a t i o n h a d a p e r i o d of a b o u t 10 m o n t h s . T h e effect of the E u l e r i a n n u t a t i o n a p p e a r s as a small v a r i a t i o n in t h e o b s e r v e r ' s position, which has been m o r e easily d e t e c t e d f r o m a s t r o n o m i c a l o b s e r v a t i o n s as a v a r i a t i o n of latitude. T h e v a r i a t i o n s in longitude a n d a z i m u t h h a v e n o t y e t b e e n m e a s u r e d directly; t h e first o b s e r v a t i o n a l results showing the v a r i a t i o n of l a t i t u d e were o b t a i n e d at t h e e n d of t h e last c e n t u r y . S. C. CHANDLER (1891, 1892), in a r e m a r k a b l e series of p a p e r s involving t h e discussion of n u m e r o u s observations, d e t e r m i n e d a period of a b o u t 14 m o n t h s for t h e v a r i a t i o n of latitude. 1
2
The Values of the l~utations a n d of the Period of the Variation of L a t i t u d e
CHA~DLER'S results were received with scepticism because they were not in agreement with Euler's ten-month period deduced theoretically. The explanation of this disagreement was given by S. N~WCOMB (1892), who pointed out t h a t the defect was due to this theory which considered the Earth as a rigid body. NEWCOMB showed qualitatively t h a t the period will be lengthened if we take account of the fluidity of the oceans and the elasticity of the Earth. The calculations of S. S. HOUGH (1895) showed t h a t the period of the Eulerian nutation would be shortened, if the inner part of the Earth was considered as fluid. T~. SLOUDSKY(1895) set up a simplified theory of the rotation of the Earth, considering a fluid core, and determined a period of 12 or 14 months for the Eulerian nutation, depending on the form of the core. The determination of the period of the variation of latitude from observation was the first set of astronomical data t h a t showed t h a t the Earth had a certain elasticity. At the time it was not possible to use quantitative theoretical results because of the lack of knowledge of the Earth's elastic constants. Geophysical observations were not yet numerous enough to yield reliable data about the Earth's interior. When we consider the disturbing forces in the equations of motion, the corresponding nutations are called forced nutations. Those of them which have the largest numerical values are: the 19-year lunar nutation (the amplitude of this nutation in obliquity is called the constant of nutation), the semi-annual solar nutation, and the fortnightly lunar nutation. The theoretical expressions of the constants of luni-solar precession P and nutation N are of the forms
P=~ C- A
N=
C
i~
(1)
7 I+/.L
where # is the ratio of the mass of the Moon to that of the Earth, A and C are the moments of inertia of the Earth, and ~, fl, 7 are accurately known constants for a certain epoch. The theoretical values of the nutations have been determined by considering the Earth as a rigid body, and by applying the principles of rigid dynamics. The observed value of the constant of nutation 9-"2100 (NEwCO~B), was employed by WOOLARD (1953) for the determination of the amount of the other nutations; his values are 0:5522 for the semi-annual nutation and 0:0884 for the fortnightly nutation in the obliquity. The determination of the nutations from astronomical observations is difficult, because they have either a long period (19-yearly nutation) or a very small amplitude (semi-annual and fortnightly nutations). The observed value which was determined most often was the value of the constant of nutation. Selecting the best observations, S. Nv.wcoMB (1895) obtained 9:2104- 0:008; using more recent observations, I-I. SPENCER JONES (1939) and H. R. MORGAN (1943) found, respectively, 9."2066 4- 0'.'0055 (this is G. M. CLEMENCE'S, 1948, corrected value) and 9:206 4- 0'007. I f we know the observed values of the constants of luni-solar precession P and nutation N, the expressions (1) permit the determination of the quantities # and
R. O. VICENTE
3
(C-A)/C,
which are important for many astronomical and geophysical investigations. From the values of the mass of the Moon and the constant of precession, it is possible to compute the constant of nutation. The value derived by SPENCEH JONES (1941) is 9'2272 +_0'0008. Comparing this with the observed value, it is seen at once t h a t there is an obvious discrepancy ; this was first emphasized by J. JACKSON (1930). 2. INTEHNAL CONSTITUTION OF THE EARTH The possibility of explaining the discordancies exhibited by the value of the constant of nutation depends on a better knowledge of the internal constitution of the E a r t h ; a theory has to be designed which adopts a model of the Earth t h a t agrees more closely with the actual Earth. The discordance between theoretical and observed values of the nutations was only detectable in the value of the 19-yearly nutation in obliquity, because the observations had not been analysed for the 19-yearly nutation in longitude, nor for the semi-annual and fortnightly nutations. The computation of the semi-annual and fortnightly nutations from observations is difficult because of their small amplitude; however, this work has been done in recent years and will be mentioned later on. These discordances, and also t h a t involving the period of the variation of latitude, indicated the need for a theory taking into account the internal constitution of the Earth. During the last century several models were considered for the distribution of density in the Earth which need not to be described in detail, because they are not based on geophysical observations. Some of them, like LAPLACE'S and ROCHE'S models, have the advantage of making the calculations easier in all the problems connected with the figure of the Earth. The model considered by WIECHERT (1897) supposes the Earth to be composed of a shell and a core, both of uniform density; this model was used in several geophysical investigations. The accumulation of geophysical data concerning the interior of the Earth, obtained especially from seismological investigations, has permitted the determination of fairly good values for the density distribution and the elastic constants in the Earth (BuLLEN, 1953). This knowledge has resulted in many advances in all problems connected with the constitution of the Earth's interior. 3. THE THEORY OF THE NUTATION The nutations are produced by different components of the bodily tide. It is convenient to know the periods of the components of the tidal potential, which give rise to the different nutations, so t h a t we can know whether it is necessary to apply a dynamical theory to the Earth model adopted, or whether it is sufficient to use a statical theory. Considering a sphere of the size and mass of the Earth, the period of free oscillation of tidal type is about 1 hour 34 min if the sphere is fluid (KELVIN, 1863), and about 1 hour 6 rain if it is as rigid as steel (LAMB, 1882). These periods are short in comparison with those of the diurnal tides which give rise to the nutations we have considered; and this applies of course also to all the components of the tidal potential with periods longer than a day.
4
The Values of the Nutations and of the Period of the Variation of Latitude
We can thus be fairly confident t h a t it is sufficient to employ a statical theory for the shell of the Earth model adopted. For a liquid core it was shown by H. JEFFREYS (1949) t h a t a statical theory is still valid for the semi-diurnal, fortnightly and semiannual tides. Another point which is necessary to appreciate is the effect t h a t the components of the tidal potential have on the position of the axis of rotation of the Earth. I f the core boundary were an exact sphere, the shell could rotate without affecting the motion of the core; but in view of the ellipticity of the core, the theory shows t h a t there is a great difference between the displacements that alter the direction of the principal axis, and those t h a t do not affect its position. The free Eulerian nutation, which gives the period of the variation of latitude, tends to alter the position of the axis of rotation, and the same applies to all diurnal tides, which include the forced nutations indicated. For this reason it is necessary to apply a dynamical theory to the motion of the liquid core. Several investigations have been made on the basis of an Earth model composed of a shell and a core. The results referring to the 19-yearly nutation show t h a t considering a rigid shell and a fluid core (POINCAR]~, 1910) the amplitude would be reduced; taking into account the elasticity of a homogeneous shell (JEFFREYS, 1949), part of the effect of the fluid core is compensated, and the result comes nearer to the observed value. BO~DI and LYTTLETON (1948, 1953) have studied the problems arising from the existence of viscosity in the core, and they have shown the difficulties t h a t appear in the investigations of the core motions. For the determination of the values of the nutations it is then necessary to consider a model of the Earth that is as much as possible in agreement with the known constitution of the Earth. At the same time, the model adopted must be such t h a t it leads to a solution of the dynamical equations of motion. The indications given above show t h a t the determination of the values of the nutations is connected with the problem of the bodily tide of the Earth. This problem has been solved with greater precision when using the new data about the Earth's interior now available, and employing numerical methods of integration. H. TAKEUCHI (1950), adopting a model of the E a r t h based on BULLEN'S values and using a statical theory, reduces the problem to the solution of a system of three differential equations of the second order, calculating the values of the bodily tide numbers h, k and I. M. S. MOLODENSKII(1953), also by numerical integration, finds values for the bodily tide numbers by adopting Earth models leading to the observed velocity of seismic waves, but having different density distributions. I n view of the fact t h a t for the problem of the numerical determination of the nutations it is sufficient to use a statical theory of the shell and a hydrodynamical theory of the core, it was possible to develop a theory satisfying these conditions. The statical theory of the shell developed by TAKEUCttI iS given in terms of six adjustable constants. This fact makes it possible to use his theory of the shell (with some modifications in the boundary conditions). In this way H. JEFFREYS and R. O. VICENTE (1957a, b) have developed a theory of the nutation, considering simplified models of the core. Since our actual knowledge of the core shows t h a t its structure and composition is complicated, the values referring to the inner core being not yet settled, two different core models were adopted; one composed of a homogeneous incompressible fluid with an additional particle at the centre (1957a), and the other with a quadratic
R,
O. VICENTE
5
distribution of density of the ROCHE type (1957b). The core models adopted are such t h a t the mass and moment of inertia obtained agree with the known values of these quantities. The theory developed adopts Lagrangian coordinates and then proceeds to evaluate the work function and the kinetic energy, with allowance for an initial hydrostatic stress. I t is possible to simplify the expression of the Lagrangian function, neglecting some terms because of the choice of coordinates made, and also because of the short period of the free vibrations in comparison with periods longer than a day, which produce the nutations. I t is obvious, from the configuration of its surface, t h a t the Earth is not in a hydrostatic state; but, nevertheless, this hypothesis facilitates the solution of the problem and it will be valid for certain regions in the interior of the Earth. Some evidence t h a t the Earth is not in a hydrostatic state is given by A. H. COOK (1958) in his study of the orbit of an artificial satellite. The advantage of considering two quite different models for the core lies in the fact t h a t we can see the effects of an incompressible fluid core, or of a core in which the variation of density is mainly due to compressibility (ROcHg'S model). We can be fairly confident t h a t the actual behaviour of the Earth's core lies between these extreme cases. The solution of the Lagrangian equations of motion shows that the roots are grouped in pairs. This feature was first noticed by POINCAR~ (1910), who emphasized the importance of this type of root in all problems concerned with precession and nutation, producing the phenomenon which P o i ~ c A ~ calls "double resonance". I f there is only resonance, t h a t is, only one root very near to the period of one of the free oscillations of the system, the system behaves nearly as a solid body, and the liquid core has no influence on the period of the oscillations. If, however, instead of one root there is a pair of roots, under the same conditions, we have double resonance; the influence of the liquid core becomes important and the amplitude of the nutations will be different in comparison with a solid body. The existence of elasticity does not alter the conclusions referring to double resonance. 4. T H E
PERIOD
OF T H E V A R I A T I O N OF L A T I T U D E
I f we do not consider the disturbing forces, the solution of the equations of motion (assuming a time-factor exp (iyt), where y is the speed of a circular motion relative to axes rotating with the Earth's mean angular velocity), gives the free motions. The values are near 0 and -co, where oJ is the speed of rotation of the Earth. The results can be summarized in the following Table 1. Table 1
Model
Values o f ~,
Eulerian Nutation
('~ly) 0
--OJ
1
Central Particle
392-4 days 0-00255 ~ 0
--oJ--0"00224 co
0'00255
--OJ
mOJ
--oJ--0.00403 co
- - ~ + 0 - 0 0 6 7 9 oJ
Roche
1
394.9 d a y s 0.00253 ~
0-00253
6
The Values of the l~utatlons and of the Period of the Variation of Latitude
The most important of the free periods indicated is the Eulerian nutation. Its values are practically the same for both models, showing t h a t the consideration of compressibility in the core does not produce an appreciable change. The other free periods correspond to relative motions of the shell and core, or to the motion of the whole body; they are not relevant to the nutations, although presenting great geophysical interest. The period of the Enlerian nutation deduced from this theory should now be compared with the values furnished by the observations. Before doing so, however, we have to take into account the influence of the ocean on the theoretical value which did not allow for the fact t h a t the Earth is covered by oceans. The effect of the oceans on the period of the Enlerian nutation can be determined in the following way (JEFFgEYS and ¥ICENTE, 1957b): it depends on the bodily tide number k for an entirely elastic body; considering the model of an E a r t h covered by a shallow ocean, we can evaluate from a set of equations the corresponding value of k and the period. The period of 392 days, determined theoretically, corresponds to a certain value of k (for an entirely elastic body). I f we introduce this value into the equations, we arrive at a free period of 449 days: thus, the lengthening due to the ocean for the model considered, i.e. one covered by a shallow ocean, is 57 days. Allowing for the fact t h a t the ocean does not cover the whole surface, we can state t h a t the effect amounts to about 38 days. The free period, determined theoretically and allowing for the existence of the oceans, is then about 430 days. The investigations concerned with the determination of the period of the free Eulerian nutation from observations have been numerous and some of the results have been criticized. The value of the free Eulerian nutation can be deduced from astronomical observations of the variation of latitude. We have already mentioned the initial determinations of CHANDLER. With the setting up of the International Latitude Service, the observations of latitude variation have increased not only in number but also in precision. From the outset it was thought t h a t the determination of the period of the variation of latitude would become easier with the increase in the number of observations. The great number of investigations made, based on the results of the International Latitude Service, have shown some of the difficulties which exist in the determination of the period. The most recent investigations, based on a large number of observations, are due to H. JEFFREYS {1940), T. NICOLINI (1950), A. DANJON and B. GuINOT (1954), and A. M. WALKER and A. YOUNO (1955, 1957). The great majority of investigations employed harmonic analysis of the observational series, but sometimes the data obtained by different authors disagree; for instance, NICOLINI'S work was criticized by DANJON and GUINOT who, with the same observations, obtained different values for the period of the variation of latitude by a different grouping of the observations. The utilization of harmonic analysis in the determination of the free period has been criticized; only the investigations of JEFFREYS (1940) and of WALKER and YOUNG (1955) are based on certain statistical models, which are valid only if the random disturbance satisfies a number of conditions. WALKER and YOUNG (1957) show clearly the disadvantages of applying harmonic analysis to this type of problem. The various results derived from the observations show a fair agreement with the above-mentioned theoretical value, in spite of all the difficulties involved.
~.
O . VICENTE
7
5. T H E FORCED NUTATIONS
The dynamical equations of motion, considering the existence of disturbing forces, give the values of the forced nutations. The disturbing forces of greatest interest correspond to applied couples with periods 7 = - o J + n, where n is small and represents the speed of a circular motion with respect to axes fixed in space. The theory developed by JEFFI~EYS and VICENTE (1957a, b) considers the displacements to be built up of several motions which include the general rotation of the shell. I t is convenient to take as a standard of comparison the value of ~ for a rigid E a r t h ; this will be designated by ~. From the solution of the equations of motion we obtain the results given in Table 2: Table 2
n
Nutation Component
60
1
6800 0 1
6800 1
183 1
13.7
Central Particle Model
Roche Model
Principal Nutation
0-9964
0"9989
Precession
1-0000
1"0000
Principal Nutation {correction)
1.0036
1"0012
Semi-annual
1.0350
0-9707
Fortnightly
1.0269
1.0266
The numerical values in Table 2 show the possible limits for the correcting factors, corresponding to the principal nutations in the cases of two different Earth models: one supposes an incompressible fluid core, and the other a compressible core; the ROCHE model overestimates the compressibility; thus the values for the actual Earth will lie between the limits indicated. We know t h a t the theoretical values of the nutations have been calculated considering the E a r t h as rigid; the ratio of the observed and theoretical values gives us therefore an indication of the behaviour of the actual Earth. This ratio can then be compared with the ratio given in Table 2. We have already mentioned the observed value 9."2066 (SPENCER JONES, 1939), as well as the computed value 9'-'2272 (SPENCER JONES, 1941) of the constant of nutation. The ratio of these quantities is 0.9978, which lies between the tabulated values. Another comparison can be made with the results derived by E. P. FEDOROV (1958a) from latitude observations mainly due to the International Latitude Service. These results have the advantage of giving not only the nutation in obliquity, but also the nutation in longitude. From the observed and theoretical values given by FEDOROV, we can determine their ratio: it is 0.9976 for the nutation in obliquity, and 0-9977 for the nutation in longitude. These figures are in agreement with the tabulated values and also with SPENCER JONES' results. This is the first time t h a t
8
The Values of the Nutations and of the Period of the Variation of Latitude
the 19-yearly nutation in longitude has been calculated from a long series of observations, showing agreement with the results known previously for the nutation in obliquity. The conclusions obtained b y F~DOROV are based on about 135,000observations for the 19-yearly nutation, and on about 230,000 observations for the fortnightly nutation; it will be many years before a better series of equal length becomes available. The details of the calculations are given in another paper (FEDoRov, 1958b). From F~.DOROV'S investigations it is also possible to calculate the ratio of the observed and theoretical values for the fortnightly nutation; the values are about 5 per cent above the tabulated values. In the case of the semi-annual nutation, the calculated ratio is in better agreement with the central particle model than with the ROCHE model, but FEDOROV remarks that the observed values are based on very few data. A detailed comparison of the theoretical and observational values of the nutations has been made by H. JEFFREYS (1959). His discussion shows that the discordances previously found have been reduced, but that there is still some disagreement between the two sets of values. A good check for any theory of the nutation, which takes into account the internal constitution of the Earth, is given by the values of the bodily tide numbers h, k and l as determined b y the theory. These values must agree with the observed values of the bodily tide numbers as determined by geophysical investigations. The comparison of the observed values with those calculated from the above-mentioned theory shows good agreement. 6. F U T U R E VISTAS
The account given of the actual state of the problem of the values of the nutations and the period of the variation of latitude points to some of the directions and possibilities in which we may expect improvements in the future. One of the needs is to have numerous series of very precise observations which will give the possibility of determining more accurately the observed values of the nutations of smaller amplitude. This is now within the actual reach of several types of instruments, for instance, of photographic zenith tubes and of Danjonastrolabes. It will be convenient if the observations are presented in such a way that it is possible to separate easily the nutations in obliquity from the nutations in longitude. Investigations of another type, which are being carried out, refer to the study of the remarkable series of observations obtained b y the International Latitude Service during the half-century of its existence, with the idea of making readily comparable observations made at different periods, i.e., to allow for changes in the programmes of observations and, furthermore, to choose the best type of numerical analysis which is applicable to the given observational series. A comparison of values of the nutations derived from observations in different latitudes is worthwhile, because most of the observations have been made at the International Latitude Service observatories situated on the parallel 39 ° N. The results obtained at different latitudes might lead to better empirical values of the quantity 1 + k - 1 which has great importance for the study of the Earth tides. On the theoretical side it is unfortunate that several authors have employed different systems of axes of reference for the dynamical equations of motion. It is
R. O. VICENTE
9
also necessary to note that in the solution of these equations different approximations have been used. The motions in the core, and the existence of the inner core, are also important for the theory of nutation, as was shown previously. Geophysical investigations of the size and constitution of the inner core will be important. The theory developed b y JEFFREYS and VICENTE does not yet allow for the size of the inner core. We should note that all previous determinations have assumed the rigid-body value for the ratios of the observed and theoretical values of the nutations. We must remember, as was said in Section 1, that the displacement of the pole in longitude affects the observations used to determine the nutations and that this effect has not yet been considered. The above results showed the disagreement between the values calculated from observations and from theory. I t was also indicated how both results could be reconciled, taking into consideration the elastic properties and the internal constitution of the Earth. Of course, the best way would be to set up a theory which considered the complete equations of motion in such a manner that it takes in all details of the constitution and elastic properties of the Earth. This would be the only method of avoiding differing values for the nutations which are nowadays calculated b y the following processes: (1) b y rigid dynamics, (2) b y employing values of other astronomical quantities, and (3) b y using Earth-models based on the present knowledge of the constitution of the Earth.
REFERENCES
BONDI, H. and LYTTLETON, R. A . . BULLEN, K . E .
.
CHANDLER, S . C .
CLEMENCE, G . M . COOK, A . H . . . . DANJON, A. a n d GUINOT, B. FEDOROV, E . P .
.
1948 1953 1953 1891 1892 1948 1958 1954 1958a
1958b
HOUGH, S. S. JACKSON, J. JEFFREYS, H. JEFFREYS, H. and VICENTE, a. 0. KELVIN, Lord LAMB,
H.
.
MOLODENSKY, M~ S. MORGAN, H. R. NEWC0MB, S.
NICOLINI, T. POINCA:R~, H. SLOUDSKY, T h . . SPENCER JONES, H..
1895 1930 1940 1949 1959 1957a 1957b 1863 1882 1953 1943 1892 1895 1950 1910 1895 1939 1941
Proc. Camb. Phil. Soc., 44, 345-359. Proc. Camb. Phil. Soc., 49, 498-515. A n Introduction to the Theory of Seismology; 2nd edition; Cambridge University Press. Astr. J., l l , 59-61, 65-70, 75-79, 83-86. Astr. J . , 12, 17-22, 57-62, 65-72, 97-101. Astr. J., 53, 169-179. Geophys. J . , 1,341-345. C. R. Acad. Sci. Paris, 238, 1081-1083. N u t a t i o n as derived from latitude observations. {Symposium on the Rotation of the Earth and Atomic Time-Standards, 10th General Assembly I.A.U., Moscow.) N u t a t i o n and the Forced Motion of the E a r t h ' s Pole, Kiev. Phil. Trans. {London), A., 186, 469-506. Mort. Not. R. astr. Soc., 90, 733-742. Mon. Not. R. astr. Soc., 100, 139-155. Mon. Not. R. astr. Soc., 109, 670 687. Mon. Not. R. astr. Soc., 119, 75-80. Mon. Not. R. astr. Soc., 117, 142-161. Mon. Not. R. astr. Soc., 117, 162-173. Phil. Trans. (London), A.153, 583-616. Proc. Lond. H a t h . Soc., 13, 189-212. Trndy, Geofiz. Inst., N.19, 3-52. Astr. J., 50, 125-126. Mon. 1Vot. R. astr. Soc., 52, 336-341. Astronomical Constants, W a s h i n g t o n . Trans. Int. astr, Un., 7, 206-210. Bull. Astronomique, 27, 321-356. Bull. Soc. I m p . 1Vatur., Moscou, No. 2. 285--318. Mon. Not. R. astr. Soc., 99, 211-6. Mort. 2qot. R. astr. Soc., 101, 356-66.
10
The Values of the l~utations and of the Period of the Variation of Latitude
TAKEUCHI, H . . . . WALKER, A. i~I. and YOUNG, A. WIECHERT, E. WOOLARD, E. Wl
1950 1955 1957 1897 1953
Trans. Amer. Geophys. Un., 31, 651-689. Mon. Not. R. astr. Soc., 115, 443-459. Mon. Not. R. astr. Soc., 117, 119-141. Nachr. Ges. Wiss., G6ttingen, 221-243. Astr. Pap., Wash. 15, P a r t I.
agreement because the theoretical calculations did not take into account the internal constitution of the Earth. A theory of the nutation is described that gives values for the nutations and the period of latitude variation in better agreement with the observations than the previous theories because it considers the elastic properties and internal constitution of the Earth. Some possible improvements in the observational and theoretical aspects of the problem are mentioned.
1. INTI~ODUCTION THE d e t e r m i n a t i o n of t h e values of t h e several n u t a t i o n s is one of those p r o b l e m s in a s t r o n o m y which offer good e x a m p l e s of a d v a n c e s m a d e in successive periods of t h e o r e t i c a l a n d of o b s e r v a t i o n a l researches. T h e o b s e r v a t i o n s led to results which were n o t e x p e c t e d theoretically, a d i s c r e p a n c y caused b y over-simplified h y p o t h e s e s . T h e n u t a t i o n due to t h e S u n ' s influence was first n o t e d b y NEWTO:N ; t h e irregularities caused b y t h e m o t i o n of t h e Moon's nodes were discovered b y BRADLEY f r o m a r e m a r k a b l e series of o b s e r v a t i o n s lasting eighteen years. T h e theoretical e x p l a n a t i o n of t h e n u t a t i o n s was g i v e n b y D'ALEMBERT, s h o r t l y a f t e r B~ADLEY'S results were published. T h e t h e o r y was p e r f e c t e d b y EULER a n d LAPLACE. Since t h e n n u m e r o u s investigations h a v e b e e n m a d e ; t h e latest results, giving full expressions for t h e n u t a t i o n s a n d t h e i r n u m e r i c a l values, were p u b l i s h e d b y E. W. WOOLARD (1953). W h e n , in t h e solution of t h e d y n a m i c a l equations of motion, we do n o t consider t h e d i s t u r b i n g forces, a n d r e g a r d t h e E a r t h as s i m p l y set in r o t a t i o n a n d left to itself, t h e result o b t a i n e d is a small oscillation of t h e E a r t h ' s axis, which is called t h e free E u l e r i a n n u t a t i o n . T h e initial theoretical researches showed t h a t this n u t a t i o n h a d a p e r i o d of a b o u t 10 m o n t h s . T h e effect of the E u l e r i a n n u t a t i o n a p p e a r s as a small v a r i a t i o n in t h e o b s e r v e r ' s position, which has been m o r e easily d e t e c t e d f r o m a s t r o n o m i c a l o b s e r v a t i o n s as a v a r i a t i o n of latitude. T h e v a r i a t i o n s in longitude a n d a z i m u t h h a v e n o t y e t b e e n m e a s u r e d directly; t h e first o b s e r v a t i o n a l results showing the v a r i a t i o n of l a t i t u d e were o b t a i n e d at t h e e n d of t h e last c e n t u r y . S. C. CHANDLER (1891, 1892), in a r e m a r k a b l e series of p a p e r s involving t h e discussion of n u m e r o u s observations, d e t e r m i n e d a period of a b o u t 14 m o n t h s for t h e v a r i a t i o n of latitude. 1
2
The Values of the l~utations a n d of the Period of the Variation of L a t i t u d e
CHA~DLER'S results were received with scepticism because they were not in agreement with Euler's ten-month period deduced theoretically. The explanation of this disagreement was given by S. N~WCOMB (1892), who pointed out t h a t the defect was due to this theory which considered the Earth as a rigid body. NEWCOMB showed qualitatively t h a t the period will be lengthened if we take account of the fluidity of the oceans and the elasticity of the Earth. The calculations of S. S. HOUGH (1895) showed t h a t the period of the Eulerian nutation would be shortened, if the inner part of the Earth was considered as fluid. T~. SLOUDSKY(1895) set up a simplified theory of the rotation of the Earth, considering a fluid core, and determined a period of 12 or 14 months for the Eulerian nutation, depending on the form of the core. The determination of the period of the variation of latitude from observation was the first set of astronomical data t h a t showed t h a t the Earth had a certain elasticity. At the time it was not possible to use quantitative theoretical results because of the lack of knowledge of the Earth's elastic constants. Geophysical observations were not yet numerous enough to yield reliable data about the Earth's interior. When we consider the disturbing forces in the equations of motion, the corresponding nutations are called forced nutations. Those of them which have the largest numerical values are: the 19-year lunar nutation (the amplitude of this nutation in obliquity is called the constant of nutation), the semi-annual solar nutation, and the fortnightly lunar nutation. The theoretical expressions of the constants of luni-solar precession P and nutation N are of the forms
P=~ C- A
N=
C
i~
(1)
7 I+/.L
where # is the ratio of the mass of the Moon to that of the Earth, A and C are the moments of inertia of the Earth, and ~, fl, 7 are accurately known constants for a certain epoch. The theoretical values of the nutations have been determined by considering the Earth as a rigid body, and by applying the principles of rigid dynamics. The observed value of the constant of nutation 9-"2100 (NEwCO~B), was employed by WOOLARD (1953) for the determination of the amount of the other nutations; his values are 0:5522 for the semi-annual nutation and 0:0884 for the fortnightly nutation in the obliquity. The determination of the nutations from astronomical observations is difficult, because they have either a long period (19-yearly nutation) or a very small amplitude (semi-annual and fortnightly nutations). The observed value which was determined most often was the value of the constant of nutation. Selecting the best observations, S. Nv.wcoMB (1895) obtained 9:2104- 0:008; using more recent observations, I-I. SPENCER JONES (1939) and H. R. MORGAN (1943) found, respectively, 9."2066 4- 0'.'0055 (this is G. M. CLEMENCE'S, 1948, corrected value) and 9:206 4- 0'007. I f we know the observed values of the constants of luni-solar precession P and nutation N, the expressions (1) permit the determination of the quantities # and
R. O. VICENTE
3
(C-A)/C,
which are important for many astronomical and geophysical investigations. From the values of the mass of the Moon and the constant of precession, it is possible to compute the constant of nutation. The value derived by SPENCEH JONES (1941) is 9'2272 +_0'0008. Comparing this with the observed value, it is seen at once t h a t there is an obvious discrepancy ; this was first emphasized by J. JACKSON (1930). 2. INTEHNAL CONSTITUTION OF THE EARTH The possibility of explaining the discordancies exhibited by the value of the constant of nutation depends on a better knowledge of the internal constitution of the E a r t h ; a theory has to be designed which adopts a model of the Earth t h a t agrees more closely with the actual Earth. The discordance between theoretical and observed values of the nutations was only detectable in the value of the 19-yearly nutation in obliquity, because the observations had not been analysed for the 19-yearly nutation in longitude, nor for the semi-annual and fortnightly nutations. The computation of the semi-annual and fortnightly nutations from observations is difficult because of their small amplitude; however, this work has been done in recent years and will be mentioned later on. These discordances, and also t h a t involving the period of the variation of latitude, indicated the need for a theory taking into account the internal constitution of the Earth. During the last century several models were considered for the distribution of density in the Earth which need not to be described in detail, because they are not based on geophysical observations. Some of them, like LAPLACE'S and ROCHE'S models, have the advantage of making the calculations easier in all the problems connected with the figure of the Earth. The model considered by WIECHERT (1897) supposes the Earth to be composed of a shell and a core, both of uniform density; this model was used in several geophysical investigations. The accumulation of geophysical data concerning the interior of the Earth, obtained especially from seismological investigations, has permitted the determination of fairly good values for the density distribution and the elastic constants in the Earth (BuLLEN, 1953). This knowledge has resulted in many advances in all problems connected with the constitution of the Earth's interior. 3. THE THEORY OF THE NUTATION The nutations are produced by different components of the bodily tide. It is convenient to know the periods of the components of the tidal potential, which give rise to the different nutations, so t h a t we can know whether it is necessary to apply a dynamical theory to the Earth model adopted, or whether it is sufficient to use a statical theory. Considering a sphere of the size and mass of the Earth, the period of free oscillation of tidal type is about 1 hour 34 min if the sphere is fluid (KELVIN, 1863), and about 1 hour 6 rain if it is as rigid as steel (LAMB, 1882). These periods are short in comparison with those of the diurnal tides which give rise to the nutations we have considered; and this applies of course also to all the components of the tidal potential with periods longer than a day.
4
The Values of the Nutations and of the Period of the Variation of Latitude
We can thus be fairly confident t h a t it is sufficient to employ a statical theory for the shell of the Earth model adopted. For a liquid core it was shown by H. JEFFREYS (1949) t h a t a statical theory is still valid for the semi-diurnal, fortnightly and semiannual tides. Another point which is necessary to appreciate is the effect t h a t the components of the tidal potential have on the position of the axis of rotation of the Earth. I f the core boundary were an exact sphere, the shell could rotate without affecting the motion of the core; but in view of the ellipticity of the core, the theory shows t h a t there is a great difference between the displacements that alter the direction of the principal axis, and those t h a t do not affect its position. The free Eulerian nutation, which gives the period of the variation of latitude, tends to alter the position of the axis of rotation, and the same applies to all diurnal tides, which include the forced nutations indicated. For this reason it is necessary to apply a dynamical theory to the motion of the liquid core. Several investigations have been made on the basis of an Earth model composed of a shell and a core. The results referring to the 19-yearly nutation show t h a t considering a rigid shell and a fluid core (POINCAR]~, 1910) the amplitude would be reduced; taking into account the elasticity of a homogeneous shell (JEFFREYS, 1949), part of the effect of the fluid core is compensated, and the result comes nearer to the observed value. BO~DI and LYTTLETON (1948, 1953) have studied the problems arising from the existence of viscosity in the core, and they have shown the difficulties t h a t appear in the investigations of the core motions. For the determination of the values of the nutations it is then necessary to consider a model of the Earth that is as much as possible in agreement with the known constitution of the Earth. At the same time, the model adopted must be such t h a t it leads to a solution of the dynamical equations of motion. The indications given above show t h a t the determination of the values of the nutations is connected with the problem of the bodily tide of the Earth. This problem has been solved with greater precision when using the new data about the Earth's interior now available, and employing numerical methods of integration. H. TAKEUCHI (1950), adopting a model of the E a r t h based on BULLEN'S values and using a statical theory, reduces the problem to the solution of a system of three differential equations of the second order, calculating the values of the bodily tide numbers h, k and I. M. S. MOLODENSKII(1953), also by numerical integration, finds values for the bodily tide numbers by adopting Earth models leading to the observed velocity of seismic waves, but having different density distributions. I n view of the fact t h a t for the problem of the numerical determination of the nutations it is sufficient to use a statical theory of the shell and a hydrodynamical theory of the core, it was possible to develop a theory satisfying these conditions. The statical theory of the shell developed by TAKEUCttI iS given in terms of six adjustable constants. This fact makes it possible to use his theory of the shell (with some modifications in the boundary conditions). In this way H. JEFFREYS and R. O. VICENTE (1957a, b) have developed a theory of the nutation, considering simplified models of the core. Since our actual knowledge of the core shows t h a t its structure and composition is complicated, the values referring to the inner core being not yet settled, two different core models were adopted; one composed of a homogeneous incompressible fluid with an additional particle at the centre (1957a), and the other with a quadratic
R,
O. VICENTE
5
distribution of density of the ROCHE type (1957b). The core models adopted are such t h a t the mass and moment of inertia obtained agree with the known values of these quantities. The theory developed adopts Lagrangian coordinates and then proceeds to evaluate the work function and the kinetic energy, with allowance for an initial hydrostatic stress. I t is possible to simplify the expression of the Lagrangian function, neglecting some terms because of the choice of coordinates made, and also because of the short period of the free vibrations in comparison with periods longer than a day, which produce the nutations. I t is obvious, from the configuration of its surface, t h a t the Earth is not in a hydrostatic state; but, nevertheless, this hypothesis facilitates the solution of the problem and it will be valid for certain regions in the interior of the Earth. Some evidence t h a t the Earth is not in a hydrostatic state is given by A. H. COOK (1958) in his study of the orbit of an artificial satellite. The advantage of considering two quite different models for the core lies in the fact t h a t we can see the effects of an incompressible fluid core, or of a core in which the variation of density is mainly due to compressibility (ROcHg'S model). We can be fairly confident t h a t the actual behaviour of the Earth's core lies between these extreme cases. The solution of the Lagrangian equations of motion shows that the roots are grouped in pairs. This feature was first noticed by POINCAR~ (1910), who emphasized the importance of this type of root in all problems concerned with precession and nutation, producing the phenomenon which P o i ~ c A ~ calls "double resonance". I f there is only resonance, t h a t is, only one root very near to the period of one of the free oscillations of the system, the system behaves nearly as a solid body, and the liquid core has no influence on the period of the oscillations. If, however, instead of one root there is a pair of roots, under the same conditions, we have double resonance; the influence of the liquid core becomes important and the amplitude of the nutations will be different in comparison with a solid body. The existence of elasticity does not alter the conclusions referring to double resonance. 4. T H E
PERIOD
OF T H E V A R I A T I O N OF L A T I T U D E
I f we do not consider the disturbing forces, the solution of the equations of motion (assuming a time-factor exp (iyt), where y is the speed of a circular motion relative to axes rotating with the Earth's mean angular velocity), gives the free motions. The values are near 0 and -co, where oJ is the speed of rotation of the Earth. The results can be summarized in the following Table 1. Table 1
Model
Values o f ~,
Eulerian Nutation
('~ly) 0
--OJ
1
Central Particle
392-4 days 0-00255 ~ 0
--oJ--0"00224 co
0'00255
--OJ
mOJ
--oJ--0.00403 co
- - ~ + 0 - 0 0 6 7 9 oJ
Roche
1
394.9 d a y s 0.00253 ~
0-00253
6
The Values of the l~utatlons and of the Period of the Variation of Latitude
The most important of the free periods indicated is the Eulerian nutation. Its values are practically the same for both models, showing t h a t the consideration of compressibility in the core does not produce an appreciable change. The other free periods correspond to relative motions of the shell and core, or to the motion of the whole body; they are not relevant to the nutations, although presenting great geophysical interest. The period of the Enlerian nutation deduced from this theory should now be compared with the values furnished by the observations. Before doing so, however, we have to take into account the influence of the ocean on the theoretical value which did not allow for the fact t h a t the Earth is covered by oceans. The effect of the oceans on the period of the Enlerian nutation can be determined in the following way (JEFFgEYS and ¥ICENTE, 1957b): it depends on the bodily tide number k for an entirely elastic body; considering the model of an E a r t h covered by a shallow ocean, we can evaluate from a set of equations the corresponding value of k and the period. The period of 392 days, determined theoretically, corresponds to a certain value of k (for an entirely elastic body). I f we introduce this value into the equations, we arrive at a free period of 449 days: thus, the lengthening due to the ocean for the model considered, i.e. one covered by a shallow ocean, is 57 days. Allowing for the fact t h a t the ocean does not cover the whole surface, we can state t h a t the effect amounts to about 38 days. The free period, determined theoretically and allowing for the existence of the oceans, is then about 430 days. The investigations concerned with the determination of the period of the free Eulerian nutation from observations have been numerous and some of the results have been criticized. The value of the free Eulerian nutation can be deduced from astronomical observations of the variation of latitude. We have already mentioned the initial determinations of CHANDLER. With the setting up of the International Latitude Service, the observations of latitude variation have increased not only in number but also in precision. From the outset it was thought t h a t the determination of the period of the variation of latitude would become easier with the increase in the number of observations. The great number of investigations made, based on the results of the International Latitude Service, have shown some of the difficulties which exist in the determination of the period. The most recent investigations, based on a large number of observations, are due to H. JEFFREYS {1940), T. NICOLINI (1950), A. DANJON and B. GuINOT (1954), and A. M. WALKER and A. YOUNO (1955, 1957). The great majority of investigations employed harmonic analysis of the observational series, but sometimes the data obtained by different authors disagree; for instance, NICOLINI'S work was criticized by DANJON and GUINOT who, with the same observations, obtained different values for the period of the variation of latitude by a different grouping of the observations. The utilization of harmonic analysis in the determination of the free period has been criticized; only the investigations of JEFFREYS (1940) and of WALKER and YOUNG (1955) are based on certain statistical models, which are valid only if the random disturbance satisfies a number of conditions. WALKER and YOUNG (1957) show clearly the disadvantages of applying harmonic analysis to this type of problem. The various results derived from the observations show a fair agreement with the above-mentioned theoretical value, in spite of all the difficulties involved.
~.
O . VICENTE
7
5. T H E FORCED NUTATIONS
The dynamical equations of motion, considering the existence of disturbing forces, give the values of the forced nutations. The disturbing forces of greatest interest correspond to applied couples with periods 7 = - o J + n, where n is small and represents the speed of a circular motion with respect to axes fixed in space. The theory developed by JEFFI~EYS and VICENTE (1957a, b) considers the displacements to be built up of several motions which include the general rotation of the shell. I t is convenient to take as a standard of comparison the value of ~ for a rigid E a r t h ; this will be designated by ~. From the solution of the equations of motion we obtain the results given in Table 2: Table 2
n
Nutation Component
60
1
6800 0 1
6800 1
183 1
13.7
Central Particle Model
Roche Model
Principal Nutation
0-9964
0"9989
Precession
1-0000
1"0000
Principal Nutation {correction)
1.0036
1"0012
Semi-annual
1.0350
0-9707
Fortnightly
1.0269
1.0266
The numerical values in Table 2 show the possible limits for the correcting factors, corresponding to the principal nutations in the cases of two different Earth models: one supposes an incompressible fluid core, and the other a compressible core; the ROCHE model overestimates the compressibility; thus the values for the actual Earth will lie between the limits indicated. We know t h a t the theoretical values of the nutations have been calculated considering the E a r t h as rigid; the ratio of the observed and theoretical values gives us therefore an indication of the behaviour of the actual Earth. This ratio can then be compared with the ratio given in Table 2. We have already mentioned the observed value 9."2066 (SPENCER JONES, 1939), as well as the computed value 9'-'2272 (SPENCER JONES, 1941) of the constant of nutation. The ratio of these quantities is 0.9978, which lies between the tabulated values. Another comparison can be made with the results derived by E. P. FEDOROV (1958a) from latitude observations mainly due to the International Latitude Service. These results have the advantage of giving not only the nutation in obliquity, but also the nutation in longitude. From the observed and theoretical values given by FEDOROV, we can determine their ratio: it is 0.9976 for the nutation in obliquity, and 0-9977 for the nutation in longitude. These figures are in agreement with the tabulated values and also with SPENCER JONES' results. This is the first time t h a t
8
The Values of the Nutations and of the Period of the Variation of Latitude
the 19-yearly nutation in longitude has been calculated from a long series of observations, showing agreement with the results known previously for the nutation in obliquity. The conclusions obtained b y F~DOROV are based on about 135,000observations for the 19-yearly nutation, and on about 230,000 observations for the fortnightly nutation; it will be many years before a better series of equal length becomes available. The details of the calculations are given in another paper (FEDoRov, 1958b). From F~.DOROV'S investigations it is also possible to calculate the ratio of the observed and theoretical values for the fortnightly nutation; the values are about 5 per cent above the tabulated values. In the case of the semi-annual nutation, the calculated ratio is in better agreement with the central particle model than with the ROCHE model, but FEDOROV remarks that the observed values are based on very few data. A detailed comparison of the theoretical and observational values of the nutations has been made by H. JEFFREYS (1959). His discussion shows that the discordances previously found have been reduced, but that there is still some disagreement between the two sets of values. A good check for any theory of the nutation, which takes into account the internal constitution of the Earth, is given by the values of the bodily tide numbers h, k and l as determined b y the theory. These values must agree with the observed values of the bodily tide numbers as determined by geophysical investigations. The comparison of the observed values with those calculated from the above-mentioned theory shows good agreement. 6. F U T U R E VISTAS
The account given of the actual state of the problem of the values of the nutations and the period of the variation of latitude points to some of the directions and possibilities in which we may expect improvements in the future. One of the needs is to have numerous series of very precise observations which will give the possibility of determining more accurately the observed values of the nutations of smaller amplitude. This is now within the actual reach of several types of instruments, for instance, of photographic zenith tubes and of Danjonastrolabes. It will be convenient if the observations are presented in such a way that it is possible to separate easily the nutations in obliquity from the nutations in longitude. Investigations of another type, which are being carried out, refer to the study of the remarkable series of observations obtained b y the International Latitude Service during the half-century of its existence, with the idea of making readily comparable observations made at different periods, i.e., to allow for changes in the programmes of observations and, furthermore, to choose the best type of numerical analysis which is applicable to the given observational series. A comparison of values of the nutations derived from observations in different latitudes is worthwhile, because most of the observations have been made at the International Latitude Service observatories situated on the parallel 39 ° N. The results obtained at different latitudes might lead to better empirical values of the quantity 1 + k - 1 which has great importance for the study of the Earth tides. On the theoretical side it is unfortunate that several authors have employed different systems of axes of reference for the dynamical equations of motion. It is
R. O. VICENTE
9
also necessary to note that in the solution of these equations different approximations have been used. The motions in the core, and the existence of the inner core, are also important for the theory of nutation, as was shown previously. Geophysical investigations of the size and constitution of the inner core will be important. The theory developed b y JEFFREYS and VICENTE does not yet allow for the size of the inner core. We should note that all previous determinations have assumed the rigid-body value for the ratios of the observed and theoretical values of the nutations. We must remember, as was said in Section 1, that the displacement of the pole in longitude affects the observations used to determine the nutations and that this effect has not yet been considered. The above results showed the disagreement between the values calculated from observations and from theory. I t was also indicated how both results could be reconciled, taking into consideration the elastic properties and the internal constitution of the Earth. Of course, the best way would be to set up a theory which considered the complete equations of motion in such a manner that it takes in all details of the constitution and elastic properties of the Earth. This would be the only method of avoiding differing values for the nutations which are nowadays calculated b y the following processes: (1) b y rigid dynamics, (2) b y employing values of other astronomical quantities, and (3) b y using Earth-models based on the present knowledge of the constitution of the Earth.
REFERENCES
BONDI, H. and LYTTLETON, R. A . . BULLEN, K . E .
.
CHANDLER, S . C .
CLEMENCE, G . M . COOK, A . H . . . . DANJON, A. a n d GUINOT, B. FEDOROV, E . P .
.
1948 1953 1953 1891 1892 1948 1958 1954 1958a
1958b
HOUGH, S. S. JACKSON, J. JEFFREYS, H. JEFFREYS, H. and VICENTE, a. 0. KELVIN, Lord LAMB,
H.
.
MOLODENSKY, M~ S. MORGAN, H. R. NEWC0MB, S.
NICOLINI, T. POINCA:R~, H. SLOUDSKY, T h . . SPENCER JONES, H..
1895 1930 1940 1949 1959 1957a 1957b 1863 1882 1953 1943 1892 1895 1950 1910 1895 1939 1941
Proc. Camb. Phil. Soc., 44, 345-359. Proc. Camb. Phil. Soc., 49, 498-515. A n Introduction to the Theory of Seismology; 2nd edition; Cambridge University Press. Astr. J., l l , 59-61, 65-70, 75-79, 83-86. Astr. J . , 12, 17-22, 57-62, 65-72, 97-101. Astr. J., 53, 169-179. Geophys. J . , 1,341-345. C. R. Acad. Sci. Paris, 238, 1081-1083. N u t a t i o n as derived from latitude observations. {Symposium on the Rotation of the Earth and Atomic Time-Standards, 10th General Assembly I.A.U., Moscow.) N u t a t i o n and the Forced Motion of the E a r t h ' s Pole, Kiev. Phil. Trans. {London), A., 186, 469-506. Mort. Not. R. astr. Soc., 90, 733-742. Mon. Not. R. astr. Soc., 100, 139-155. Mon. Not. R. astr. Soc., 109, 670 687. Mon. Not. R. astr. Soc., 119, 75-80. Mon. Not. R. astr. Soc., 117, 142-161. Mon. Not. R. astr. Soc., 117, 162-173. Phil. Trans. (London), A.153, 583-616. Proc. Lond. H a t h . Soc., 13, 189-212. Trndy, Geofiz. Inst., N.19, 3-52. Astr. J., 50, 125-126. Mon. 1Vot. R. astr. Soc., 52, 336-341. Astronomical Constants, W a s h i n g t o n . Trans. Int. astr, Un., 7, 206-210. Bull. Astronomique, 27, 321-356. Bull. Soc. I m p . 1Vatur., Moscou, No. 2. 285--318. Mon. Not. R. astr. Soc., 99, 211-6. Mort. 2qot. R. astr. Soc., 101, 356-66.
10
The Values of the l~utations and of the Period of the Variation of Latitude
TAKEUCHI, H . . . . WALKER, A. i~I. and YOUNG, A. WIECHERT, E. WOOLARD, E. Wl
1950 1955 1957 1897 1953
Trans. Amer. Geophys. Un., 31, 651-689. Mon. Not. R. astr. Soc., 115, 443-459. Mon. Not. R. astr. Soc., 117, 119-141. Nachr. Ges. Wiss., G6ttingen, 221-243. Astr. Pap., Wash. 15, P a r t I.