Geophys. J. R.astr. SOC.
(1975) 43, 573-588
Total Effect of any Nearly-Diurnal Wobble of the Earth’s
Axis of Rotation in Latitude and Time Observations.
Application to the Paris Astrolabe Observations from
1956.6 to 1973.8
Nicole Capitaine
(Received 1975 January 6)
Summary
The existence of a non-spherical ideal liquid core in the Earth’s interior
produces a free nearly-diurnal wobble of the Earth’s axis of rotation. The effect of this wobble in latitude and time observations is
more complex than has generally been considered before. In previous
discussions only the variation of latitude and time due to the neglected
wobble has been considered and not the errors arising from the complementary neglected nutation in space. We establish here the theoretical
expression for the total effect and we show that the first effect is negligible
in comparison with the second. This expression is compared with similar
ones due to the diurnal wobble and the corresponding main terms of
nutation in space.
As an application we search for such expressions in the Paris astrolabe
observations from 1956.6 to 1973.8 using several kinds of analysis. We
first evaluate the amplitudes of errors in the main terms of nutation and
secondly we show that the amplitude of a free nearly-diurnal wobble, if it
exists, is at least 100 times lower than the one claimed by several authors.
Introduction
The existence of a retrograde nearly-diurnal wobble of the axis of rotation with
respect to the axis of figure, produced by the presence of a non-spherical ideal liquid
core in the Earth‘s interior, have been shown theoretically by Hough (1895), Poincark
(1910), Jeffreys (1949, 1950), Jeffreys & Vicente (1957, 1964), Molodensky (1961).
Popov (1963), Thomas (1964), DCbarbat (1969, 1971), Sugawa & Ooe (1970), Yatskiv
(1972) examined latitude and time observations and suggested some evidence for the
existence of this theoretical wobble. But Toomre stressed in 1973 (Rochester 1973;
Rochester, Jensen & Smylie 1974; Toomre 1974) that such a nearly-diurnal wobble
should induce a long-period nutation of the axis of rotation in space with a much
larger amplitude.
As we have previously suggested (Capitaine 1974), such a neglected nutation in
space would induce errors in latitude and time observations since the equatorial
co-ordinates of stars are assumed error-free in the reduction of these observational
data. We examine here the formulation of these errors, whichare similar to those due to
errors in the main terms of nutation. We compare their theoretical amplitudes with
the theoretical variations of latitude and time due to the nearly-diurnal wobble and
we study the different ways in which these errors appear.
573
T
574
N. Capitaine
1. Total effect of the nearly-diurnal wobble in latitude and time observations
1 . Variation of latitude and time
We consider a retrograde nearly-diurnal wobble of amplitude y and frequency kw,
with k nearly equal to - 1, o being the angular velocity of the Earth. In the case of
the wobble induced by the liquid core, 1 + k is the dynamical ellipticity of this core
(Molodensky 1961); it is about 1/400 (Jeffreys & Vicente 1957). We take the principal
axes of inertia GM, GN, GO, as terrestrial reference axes (Fig. I), we denote by
(wl, w2, 03)
and by (ml, m2, m3)respectively the components and the direction
cosines of the rotation axis relative to them and by to the initial epoch. At instant, f,
this wobble is such that:
\
m, = w , / o = y cos [ k o ( t - to)-o]
m2 = w 2 / o = ysin [kw(t-to)-o]
(1)
(Woolard 1953)
It gives variations d , q5 and d , C of latitude and time at a station of latitude
longitude Lo:
d14 = YCOS [kw(t-to)-o+L,]
d,C
=
9, and
-y tgq5,sin [kw(t-t,)-o+L,]
We call these variations the ' direct effect ' of the wobble in the observations.
FIG.1. Definition of the Euler's angles. G is the Earth's centre of mass.
575
Nearly-diurnal wobble of the Earth's axis
2. Errors arising from the complementary nutation in space
We examine here how the nearly-diurnal wobble produces errors in latitude and
time data through its complementary neglected nutation in space.
(a) Forced motion of the axis of maximum inertia in space. Every wobble of the
rotation axis induces a forced nutation of the axis of maximum inertia in space which
is obtained by using Euler's kinematical equations:
(
)=(-sin@
$s;e
-COSCD
o
(3)
0
&+$cose
0
Jn these equations the wobble is expressed by its components ol,
w 2 , 0,and the
nutation in spase by the temporal variations of 8, CD defined by the Fig. 1. Following
Woolard (1953), 8 is the celestial co-latitude 20 of the pole of inertia 0, the longitude
of its descending node measured in the fixed ecliptic, GXY, eastward from the fixed
equinox GX and Q the Earth's diurnal rotation.
The forced nutation corresponding to the nearly-diurnal wobble defined by (1) is
therefore such that:
1) sin 8 = - o
y sin [@ k o ( t - to)- a ]
+,
+
+
# g = - wy cos [CD +k o ( t - to)- a]
6 = CO,-~)
cod.
Variations A1 1) and A, 8 in longitude and obliquity due to this forced motion in space
and such that $ =
A1 $ and 9 = 8, + A1 8 are obtained by integrating (4) between
instants to (the chosen origin) and t.
For this purpose some simplifications are possible. We first see that A l e is
periodic with an amplitude my/(& + kw) which is certainly much less than lo-, rad, and
we know that 0, is about 23"27', so a simplification consists of neglecting cos 8, A1 0
and sin 8, A, 9 respectively in comparison with sin 8, and cos 0,. Hence:
+, +
r
t
f$ sin 8dt =
to
and
s
$(sin 8, +cos 8, A1 8)dt = sin 8, A, $,
fo
t
/$cosOdt = cost?, A,+.
f0
The second simplification consists of neglecting $ cos 8 in comparison with (I +k)m,
since the larger term of $ cos 8 due to precession is 50"/year or lop7rad/day, and
(1 + k ) o , is of the order of 2741 + k ) rad/day, that is lo-, or
rad/day. We can
then put 6 + k w = (1 + k ) o and obtain an amplitude of y/(l + k ) for A1 8 and
sin @Al+. These simplifications give the following results:
CD
= o,(t-t,)-+cose+~~
sin 86, $ = [ y / ( I
+ k ) ]cos [(I +k)w,(t- to)- /3]
A l e = -[y/(l+k)lsin [(1+k)w,(t-t0)-/3]
where
p = a++cose-mo
(5)
I
(b) Forced motion of the rotation axis in space. The forced nutation of the rotation
axis in space due to the wobble is obtained by adding the two preceding motions: the
motion of the rotation axis with respect to the axis of maximuni inertia and the motion
of the axis of maximum inertia in space.
576
N. Capitaine
The variations Alq and A 1 in
~ longitude and obliquity referred to the rotation
axis are then such that: A l q = 6,++A,+ and A,E = 6,8+A18 where sin 86,$ and
6,O are the components of the first motion respectively along the equinox line and its
east-perpendicular lying in the equatorial plane.
The nearly-diurnal wobble defined by (1) is such that:
siaO6,+ = -ycos [(l+k)03(t-to)-/3]
6,O = ysin [ ( l + k ) ~ ~ ( t - t ~ ) - P ]
and the corresponding forced nutation of the rotation axis in space can hence be
expressed as:
sinOA1q = [-k/(l +k)]ycos [(l+k)o,(t-to)-P1
(9)
A , E = [k/(l+k)]ysin [(1+k)03(t-to)-P1
)
We denote by A = ky/(l + k ) the amplitude of this nutation in space, by r = (1 + k ) 0 3
its frequency and by P2 = 1/(1+k) its period expressed in sidereal days.
(c) Corresponding variations in equatorial co-ordinates of a star. Variations Aa
and A6 of right ascension a and declinatiorl 6 of a star due to a nutation of the rotation
axis in space defined by variations Aq and AE in longitude and obliquity are expressed
as (Danjon 1959):
AG! = Aq(cos E + sin E sin a tg6) - AEtg6 cos a
Ad
Then, the variations A,
(9) are:
A,cc =
=
AEsinrx+AqsinEcosci.
o[
and A, 6 corresponding to the forced nutation defined by
- A{(cotg &+sinct tg6) cos [r(t-to)-
+
P] tg6 cos c1 sin[r(t- to)- PI}
A16 = A(sinc1sin [ ~ ( t - t , ) - ~ ] - c o s a c o s [r(t-t,)-j]}.
1
(11)
(d) Corresponding errors in latitude and time data. Erroneous equatorial COordinates of stars used in the reduction of observations introduce errors in the computed latitude and time which are the same for each star observed at the same station
(DCbarbat & Guinot 1970). They are hence equal to the errors obtained for zenith
stars with ct = T, 6 = $o ( T being the local sidereal time and 4o the latitude of the
station).
The errors Al$ and A,C deduced from (11) are:
A,$ = - ~ c o s[~+r(t-t,)-pl
+
AIC = -Atg$, sin [T r ( t - to) - PI-A cotg E cos [r(t- to)-
P]
]
(12)
We call these errors the ' indirect effect ' of the nearly-diurnal wobble as they appear
indirectly through the errors in equatorial co-ordinates of stars.
This ' indirect effect ' is R = k/(l + k ) times larger than the ' direct effect ' and its
period if P2 if the observations are always made at the same local sidereal time. The
coefficient R and the period P, increase as k approaches -1, which means as the
period of the nearly-diurnal wobble approaches a sidereal day.
2. Total effect of the main terms of nutation in latitude and time observations
Every term, i , of the luni-solar nutation in space of frequency T i = (1 + k , ) o
induces a forced wobble of frequency k i o in the Earth as appears by inverting
relation (3) (Woolard 1953). If N i and O i are respectively the coefficients in longitude
577
Nearly-diurnal wobble of the Earth’s axis
and obliquity of the term, i, the amplitude of the induced wobble is y i = A d i ( l + k i ) ,
where d i = (Ni+Oi)/2 and A is the Fedorov’s coefficient which takes into account
elasticity of the Earth (Fedorov 1963; Melchior 1973). The total forced wobble
induced by the luni-solar precession and nutation is called the ‘ diurnal wobble ’ as its
main term, has a period of one sidereal day; its amplitude is less than 0”.01 (Fedorov
1963).
In the same way as in Section 1 we have expressed the ‘ direct ’ and the ‘ indirect
effect ’ of the nearly-diurnal wobble in observations, we can express:
(i) The variations d2# and d2C which appear in latitude and time observations if
the correction of the ‘ diurnal wobble ’ is neglected.
At a station of latitude 4, and longitude Lo, they are:
a24
= COJiCOS [kio(t-t,)-oi+L,]}
i
I
d2C = -C{yitg4,sin [ k i o ( t - t o ) - o i + L , ] }
i
where k i is nearly equal to - 1 and oi is a constant.
(ii) The errors A2# and A2C induced in latitude and time data by the ‘ indirect
effect’ of errors A2a and A26 in the co-ordinates of stars used in the reduction of
observations and due to errors N , and Ri in nutation coefficients N i and Oi.
As the errors in the nutation in space corresponding to the errors N i and SZi are
such that:
sin&A2+= -C{N,sin [ r i ( t - t o ) - Z i ] }
1
i
A ~=
E - C { R ~ C O [Sr i ( t - t o ) - T i ] }
i
these errors A2 # and A2 C are for a local sidereal time T at a station of latitude + o :
A2+ = C{-Nicos [Ti(t-to)-Bi]cos T+SZisin [Ti(t-to)-fii]sin T }
i
A,C = -C{(Nicotge+Nitg+,sin
i
with
T)cos [ri(t-to)-j?,]
}
(15)
+SZi tg+,sin [Ti(t-to)-j?i]cos T }
pi = zi+ 4 2 .
If we put Ai = ( N i + QJ/2 and A - = ( N i - Ri)/2:
A24
=
-C{Aicos [ T i ( t - t o ) + T - j ? i ] }
i
A2C = -C(Nicotg&cos [Ti(t-to)-Bi}
i
-C(Aitg4,sin
i
[Ti(t-ro)+ T - S i ] }
The amplitudes yi and A, are not related and can be of the same order.
3. Observed effects in classical latitude and time data
Comparing expressions (13) and (16) to (2) and (12), we see that the formulations
of the effects considered in Sections 1 and 2 are similar, so we group them by the same
notations, putting index i, for the particular case considered in Section 1. We have
then: Ai,= N i , and A i o = Ri, yio. We denote for each term, i :
(i) By Po, = l/ki, the period of the wobble of amplitude yi expressed in sidereal
days and by PI,, this period expressed in mean days.
578
N. Capitaine
(ii) By Pz, = 1/(1 + k i ) , the period of the corresponding nutation in space
expressed in sidereal days and by PI2, this period expressed in mean days. Two
different kinds of observations can be made: the first consists of observing at a
constant mean universal time as is the case with astrolabe and meridian instrument;
the second consists of observing at a constant sidereal time as is the case for observations of the same zenith star.
The effects for these two kinds of observations are different.
1. Observations at a constant mean universal time
In this case the observations have a period of one mean solar day (md) and the
local sidereal time T has a period of 365.25 md, then:
(i) The ‘direct effect ’ appears in the observations with the amplitude y i and the
period Pl, alias of the period PIo, or PI, = Y o ,J(l+ P’o, i)md.
(ii) The ‘ indirect effect ’ of argument r i ( t - to)+ T appears in the observations
with the amplitude Ai and the period P3, = (P’2, x 365.25)/(P’,, + 365.25) md.
2. Observations at a constant sidereal time
In this case the observations have a period of one sidereal day (sd) and the local
sidereal time is constant, then:
(i) The ‘ direct effect ’ appears in the observations with the amplitude y i and the
period P4, alias of the period Po, or P4, = Po,J(l +Po, ,) = 1/(1t - k , )sd and
therefore P4, = Pz,i .
(ii) The ‘ indirect effect ’ appears in the observations with the ampitude A i and
the same period P 2 , i.
We give in Table 1, for various values of ki:
(i) The periods Po, (in sidereal hours), PI, i ,
i , P3, I (in mean days).
(ii) The ratio R i= k i / ( l+ k i ) of amplitudes of the nutation, i, in space and of its
corresponding wobble in the Earth.
The notations Np,S1 and
PI and
r1 and El correspond to the main
nutations in space and JV I, JV 11, JV 111, Mol I, Mol I1 to theoretical models of the
nearly-diurnal wobble respectively due to Jeffreys and Vicente and to Molodensky.
In the cases considered here Ri is very large since ki is nearly equal to - 1.
In the case of the nearly-diurnal wobble, Ri, is also the ratio A,,/yio of amplitudes
of the ‘ indirect ’ and ‘ direct ’ effects in the observations. The direct effect is then
negligible in comparison with the indirect effect.
4. Search for these different effects in the Paris astrolabe observations
1, Method of analysis
We have used latitude and time observations of the Paris astrolabe from 1956.6
t o 1973.8, that is 5347 values of each kind.
Each pair of values, $ I , in latitude, CI, in time is given by observation of a
group, I, of stars: 12 groups are observed during the year and we can consider that
observations are made at a constant mean universal time (UT).
Observing instruments, star programme, observing techniques and computation
method are described by Arbey & Guinot (1961), Billaud (1970), Dtbarbat & Guinot
(1970), Billaud & Guinot (1971) and Billaud (1973).
These data are homogeneous in latitude and we have homogenized them in time
bj- referring them to International Atomic Time TAI.
579
Nearlydiunal wobble of the Earth's axis
Table 1
Values of the periods Po,i,
n1
J V I11
PI
s1
NP
NP
Mol I
Mol I1
JV I
*1
J V I1
41
81
and of the ratio Ricorresponding to various
P'2,i,
values of the coeficient ki.
PO, in
sidereal
hours
-24.2418
-24.1982
-24'1643
-24.1606
-24.1 3 18
-24.1203
-24.0961
-24.0800
-24.0686
-24.0657
-24.0480
-24.0342
-24.0240
-24.0120
-24.0035
-23.9965
-23.9880
-23.9761
-23.9659
-23.9522
-23.9487
-23.9482
-23.9464
-23'9403
-23.9347
-23.9318
-23.9205
-23.9046
-23'9037
-23.8809
-23.8697
-23.8415
-23.8050
-23.7630
kr
-0.99002730
-0.99180883
-0.99320000
-0.99335154
-0.9945391 3
-0.99501365
-0-99601092
-0.99667577
-0.99715066
-0'99726970
-0.99800546
-0.99857533
-0.99900273
-0.999501 37
-0.99985331
-1.00014669
-1.00049863
- 1.00099727
-1.00142467
- 1.00199454
- 1'00214057
-1'00216367
-1~00224000
-1.00249317
- 1.00273030
- 1.00284934
- 1 -00332423
- 1.00398908
- 1.00403000
-1e00498635
- 1.00546087
- 1.00664846
-1.008191 17
- 1.00997270
P I . 1 in
mean
days
138
183
245
255
365
442
792
1679
8387
a,
-1355
-764
-575
-447
-386
-347
-309
-268
-240
-21 1
-205
-204
-201
-191
-183
- 179.
-165
-148
-148
-129
-122
- 106
-91
-79
Pf2,,in
mean
days
100
122
147
150
183
200
250
300
350
365
500
700
1000
2000
6798
-6798
-2000
- 1000
-700
-500
-466
-461
-445
-400
-365
-350
-300
-250
-247
-200
-183
-150
- 122
- 100
in
mean
days
79
91
105
106
122
129
148
165
179
183
21 1
240
268
309
347
386
447
575
764
1355
1691
1760
2034
4204
P3. I
a,
-8383
- 1679
-792
-767
-442
-365
-255
-183
-138
R,
-99
-121
- 146
-149
-182
-200
-250
-300
-350
-365
-500
-701
-1002
-2004
-6816
6818
2006
1004
703
502
468
463
447
402
367
352
302
252
249
202
184
151
123
101
Table 2
Computed values of the corrections to be added to the observed group-diflerences.
(Origin of time is 1965.0 of Besselian year)
Latitude in
10-3"
Pair
1-2
2- 3
3 4
4-5
5-6
6-7
7-8
8-9
9-10
10-11
11-12
12-1
aol.r+l
+
-48 5
-22+6
-34+7
+53+6
-3855
+66+6
-137+5
+look6
+23 6
+5055
O+ 6
+17+6
+
Time in
10-4s
boi,i+1
+70+5
-5527
+07+6
-14+7
+04+6
-15+6
+37+5
-12f6
+02?6
-91 k 6
-16i6
+22+6
Latitude in
10- '"/year
alr*l+1
-3+1
+1+1
+I51
-l+l
+1+1
+1+1
-4+ 1
+ 4+ 1
-1+1
-2+ 1
+3+ 1
+ 2+ 1
Time in
slyear
b'r.r+ 1
+3+ 1
+l+l
-151
0,l
+2+ 1
-1+1
+4+ 1
-151
+2, 1
-2+ 1
-1+1
-2* 1
580
N. Capitaine
Each pair 4,. CI, of a group I at instant t has first to be corrected for constant
and secular group corrections respectively denoted by ao, I and a l, in latitude and
by bo,I and bl, I in time.
Then, we look for terms such that:
,
f,
+CAicos
i
i
+C A i tgq50 sin
i
[ 27r(tP3,
+C N,cotgE cos
i
[ 274tP3.- to)
to)
- pi]
i
2n(t - to)
- pi]
where t is expressed in mean days, $t, and Ct, are the instantaneous values at
instant t of latitude 4 and time (TUO-TAI), minus the variations due to the nearlydiurnal and diurnal wobbles. All the terms contained in the second member of (17),
should appear in the spectral analysis of individual groups. To remove real variations
of latitude and time and make evident only terms of amplitude A i (or A i tg40) such
a n analysis can be made for group-differences between two consecutive groups
observed at 2-hr intervals. However, such a spectral analysis does not distinguish
terms due t o errors in the nutation coefficients and terms due to the neglected nearlydiurnal wobble except when the periods P l , i, P2, i, P3, (Table 1) corresponding to
the principal nutation terms are known.
In order to make a better distinction we can find from the differences, by the
method of least-squares, terms given by (15) and consider that for the nearly-diurnal
wobble we should have Ni,= Qio. All these analyses are described together with their
results in the following sections.
2. Analysis of the individual groups
Group corrections with secular variations are first applied to individual values of
+ and C. Mean Chandlerian, annual and semi-annual terms are then removed. After-
wards they are smoothed and interpolated to obtain equally spaced values which are
submitted to spectral analysis.
(a) Corrections used
(1) We have used the method of differences (Archinard 1968; DCbarbat & Guinot
1970) for computing the group corrections u0,,,bo,, and their secular variations
ul, , bl, added to observations to refer them to the mean group. For this purpose
we have first adjusted constant and secular terms among observations of every pair
of groups for all the period from 1956.5 to 1973.8. We have thus obtained the groupdifferences corrections and their secular variations denoted by a',, I + 1, boI,I + and
u ' , , , + ~ , bl,,,+l. Their values are given in Table 2. Then, we have computed
u,,,, bo,,, al, I , bl, I from the preceding values through the classical formula of the
method of differences and thus obtained the values given in Table 3.
( 2 ) Corrections used to remove the Chandlerian wobble from observations are
obtained by interpolation of amplitude (AM) and phase ( P H ) of this wobble published
,
58 1
Neariydiurnal wobble of the Earth’s axis
Table 3
Computed values of the group corrections. (Origin of time is 1965.0 of Besselian year)
Group
number
1
2
3
4
5
6
7
8
9
10
11
12
Latitude
Secular term Constant term
a , in
a, in
-2.Of 1
-66.0f6
+1.0+ 1
-16.0f 6
0.Of 1
+9.0k 6
-l.O+ 1
+45.0+6
+0.3+1 .
-5.Ok6
-0.5f 1
+35.0+6
-1.5f 1
-28.026
+111.0+6
+3.0+1
-1.Of1
+14.0+6
0.Of 1
-7.Of6
+2.Ofl
-54.0f6
+0.5 f 1
-52’0f6
Group
numbcr
1
2
3
4
5
6
7
8
9
10
11
12
Time
Secular term
bl in
s
+3.0+ 1
+0.3 k 1
-0.4 f 1
+1.0+1
+1.2f 1
-0.4f 1
fl-02 1
-2.8 f 1
-1.4f1
-3.0-1 1
-0.8 & 1
+0.6f 1
Constant term
bo in
s
+35.0f 6
-4O.Of6
t9.8f6
-2’2f6
+6.6k6
-23 k6
+75+6
-34.5 k 6
-27.7 f 6
-34’8f6
+51.0f6
t62.0f 6
by Guinot (1972) for every 0.6 years. These corrections are in latitude and time:
A M ( t ) sin {27~[t-PH(t)]/l-l90+L~)
and
+
AM(t)tg4o cos {27~[t- PH(t)]/l*190 Lo)/15
where t is the date of observation in years and AM(t), P H ( t ) the interpolated pole
co-ordinates.
(3) The annual and semi-annual terms removed are the values derived from the
observations by the method of least-squares.
They are respectively:
A S G 5= 0”.09, A , , , = 0”.02 in latitude
and
Al,, = 0s-0275, A , , , = Os.0071 in time.
(b) Smoothing and interpolation of the data. We have smoothed these 5347
corrected latitude and time values by the Vondrak’s method (Vondrak 1969) to
remove short oscillations and also very long oscillations in time which would give
noise in the spectral analysis.
These smoothed values have been interpolated to obtain a point every five days,
that is 1261 data points.
(c) Spectral analysis. These corrected, smoothed and equally spaced latitude and
time values have been submitted to spectral analysis. Computation of spectral density
S(5) for frequency 5 has been made by Blackmann and Tukey’s method (Blackmann &
Tukey 1958) using the ‘hanning’ spectral window for smoothing the spectra.
Fig. 2(a) and (b) give spectra obtained respectively in latitude and time.
We note the presence in the two spectra of peaks corresponding to the following
periods (with sometimes a slight deviation in latitude and time): 86 d, 108 d, 121 d,
200 d, 340 d and 390 d. In addition we note the peaks corresponding to 96 d, 170 d,
290 d, 450 d and 875 d in latitude and to 103 d, 185 d, 242 d, 787 d in time. Some of
these peaks can be due to real variations of latitude and time due either to polar
motion or to local phenomena as is probably the case for the 450d-peak, the
Chandlerian term having not been completely removed. Other peaks may be due to
errors in the nutation corrections as is possibly the case of the 121 d peak (corre-
N. Capitaine
582
345d
n
I
170 d
16 d
t
I
I
I
I
0.001905
0.000 194
FIG. 2(a).
a003016
Frequency ( cycles per day )
Smoothed spectrum of individual latitude values. (b) Smoothed
spectrum of individual time values.
Nearly-diurnal wobble of the Earth's axis
583
sponding to the semi-annual nutation), of the 185d peak (corresponding in time to
the semi-annual nutation at the period P2,J and of the 340d and 390d peaks
(corresponding to the 18.6 yearly nutation).
3. Analysis of the group-differences
2772 pairs of group-differences d$I, I + dC,, I + are obtained for all observations
of pairs of consecutive groups Z and Z + 1 during the period, the isolated observations
being rejected. The differences are corrected for group-differences corrections
b',, I + 1 and their secular variations a'[, I + l , b'r,
they are then smoothed
a',,
by Vondrak's method and interpolated to obtain a point every five days. The noise
level so obtained is greater than among the individual values since the number of
values used is smaller.
The corrected, smoothed and interpolated data are then used in two different
ways:
(i) in submitting them to spectral analysis by the same method as in Section 4 . 2
(ii) in deriving, by the method of Ieast-squares, the variations given by (15).
',
(a) Spectral analysis. Consecutive groups follow each other at 2-hr intervals, so
real variations of latitude and time, except the very short-period terms, vanish in
group-differences and only terms given by (12) and (16) should appear.
These terms can be written, if we put T A = TI- T,+, and TB = T,+ TI+,, TI
being the local sidereal time when the group Z is observed.
A($1-$1+1)
=
2sin TAC{Aisin[ T i ( t - t o ) + T B - D i ] )
i
A(Cr-Cr+i) = - 2 sin TA C { A i tg$, cos [ r j ( t - r o ) + T B - p i ] )
i
}
(19)
TA = -15", so 2 sin TA = -0.52.
Therefore, every term of amplitude A i , frequency Ti and phase pi appearing in
individual group analysis and corresponding to an error in the position of the rotation
axis in space should be found with the same frequency Ti, amplitude 0.52 A j and
phase n/2-fli. Results of the spectral analysis by Blackmann & Tukey's method
using the ' hanning ' spectral window are given in Fig. 3(a) and (b).
The noise level is greater than in the preceding spectra but we notice that some
peaks are found here again with sometimes a slight change in period from the former
or between latitude and time. Such is the case for peaks corresponding to periods of
108 d, 120 d, 340 d and 390 d (the last ones being separated in latitude differences only
in the unsmoothed spectrum and with a narrower step in frequency). Some peaks
found in one of the preceding spectra are found again here in the two spectra: there
is a peak corresponding to the period of 96 d which appeared in latitude, and a peak
corresponding to the period of 145 d which appeared in time. A peak is considered
as interesting if it appears in at least three analyses.
(b) Least-squares jit of periodic terms Corresponding to errors in the position of
the rotation axis in space. Theoretical terms given by (15) can be written for groupdifferences
+Qi(sin Tr-sin TI+,) sin [ T i ( t - t o ) Ai(CI-Cr+i)
=
{-Ni(sin T'-sin
pi]
T ~ + , ) c o [sT j ( t - t o ) - / ? j ]
-Qi(cos T,-cos TI+,) sin [ri(t-to)-~i}tg$o
1
I
(20)
584
N. Capitaine
-
390d
.->
n
1
c
*
FI
(b)
345 d
-E
ZOO d
L
0
a
01
VI
10
-
2864
22 5 d
90 d
149d
L
u.c31 SD5
0.000 734
FIG. 3(a).
Frequency ( cycles per day )
0.003 016
Smoothed spectrum of latitude group-differences. (b) Smoothed
spectrum of time group-differences.
Or:
Ai(41-+1+ 1)
= Xi B , sin
2n(t - to)
'2,
- Yi B , cos
i
-2, B, cos
Ai(CI-Cc,+I)/tg+o = - X i B , sin
2n(t-to)
Pz,i
2n(t- to)
-zi B , cos
Pz,i
2n(t-tJ
Pz, i
2n(t - t o )
Pz, i
- WiB,
sin
+ Yi B, cos
2n(t-t0)
Pz,i
2n(t - to)
- W iB , sin
Pz,i
27r(t-f0)
where
sin TI-sin T,,,
B2 = cos TI-COSTI+,
Yi = R i sin Pi
X i = sZi cos p i ,
W i= N i sin P i
Z i= N i cos pi,
ri = 2 n / ~ 2i .,
B,
=
Pz, i
!
I
I
585
Nearlydiurnal wobble of the Earth’s axis
Table 4
Amplitudes and periods of the main terms possibly due to errors in the position of the
rotation axis in space
Latitude: ( + c - + c + l )
Amp 1 A m p 2
in
P’z,i in
mean
days
13
11
12
11
106
157
9
10
10
18
164
183
8
10
11
13
25
11
10
8
12
24
208
237
266
284
366
9
13
4
17
9
21
432
514
18.6yr
Time: (C,-C,+,)tg+
Amp 1 Amp 2
in
s
P ‘ z , rin
mean
days
P 3 , in
mean
days
PB.-, in
82
110
111
113
122
125
132
144
154
160
182
190
198
214
347
-150
-275
-284
-298
-366
-394
-484
-671
-980
-1286
co
4782
2350
1250
386
12
9
106
13
11
160
14
19
190
15
12
13
11
11
11
283
365
395
4
24
18.6yr
mean
days
By a least-squares fit of a term given by (21), corresponding to a chosen period P 2 ,
among the values of (q51-41+ and ( C I - C I +I)/tgq50we can compute Xi, q, Zi, Wi,
that is:
Amp 1 = In,l = J(Xiz+ Y,2), Amp 2 = [ N i l = ,/(Ziz+ Wi2),
pi = Arctg(Yi/Xi) = Arctg
(2 ;)
-- -
.
We have made these computations for each period P r 2 , between 100 days and 20
years with a chosen step. We give in Table 4 the periods PIz, and their corresponding
amplitudes Amp 1 and Amp 2 for which these amplitudes are both large. We note in
the right column the corresponding periods P3, and P3+- i.
We can deduce from this that there are errors in the corrections used for the semiannual, annual and 18.6-yr nutations as they appear in the analysis through the
amplitudes Amp 1 = Ini/= lAi-A-il and Amp2 = lNil = I A , + A - , / ; they also
appeared in the spectral analysis through the coefficients A iand A - i .
The different ways in which the nutation reveals itself explains why the annual
nutation for which the error found here is such that [nil= lNil = 0’.02 (consistent
with A i= 0 (with the 183.2d period) and A V i= 0”*02,with an infinite period), did
not appear in the spectral analysis and, on the other hand, that the 18.6-yr nutation,
for which the error found here is such that Ini(
= 0.”004 and /Nil = O”702, appeared
with the same order at the two periods: P 3 , = 346 d and PJ,- = 386 d.
Assuming that no other kind of error appears with the form (21), we can deduce
the following errors in the corrections used for the main terms of nutation:
lQil = 0”~010+0”~005, (Nil = 0”~018t_Or’~O05
in 4 at 183 d
/ail= 0”-020f0“-005, lNil = 0”~020+0”~005
in C and at 366 d
Ini/= 0”~004f0”~005,INi/ = 0 ” ~ 0 2 0 ~ 0 ” ~ 0in0 54 and C at 18.6 yr.
586
N. Capitaine
On the other hand if the nearly-diurnal wobble produces an important nutation in
space it should be such that Nio = a,, so this nutation could have, from Table 4, a
period P 2 , equal to:
k 106 d, Ifr 160 d or Ifr284d
This would correspond to periods P3, respectively: 82 d or - 150 d, 110 d or - 290 d,
160d or -1286d.
Peaks corresponding to the 108 d period exist in both spectral analyses. In time,
where the period P r 2 , must also appear, a peak exists at 160 d. But a definite conclusion is difficult as we do not know a priori the period either in the Earth or in space,
of the effect for which we are looking.
5. Interpretation of the results
1. Main nutations in space
The spectral analyses presented here show that some error exists in the correction
for the principal term of the 18-6-yrnutation. This conclusion is confirmed by finding
the term of the form (21) in the group-differences at the 18.6-yr period.
However, the spectral analysis would be disturbed by the presence of an annual
term, and since we have only 17 years of data at one observatory we cannot be very
confident of the amplitude found for this error. We have found that it was five times
greater in longitude (O”02) than in obliquity (O”404) whereas other authors (Fedorov
1959; Taradii 1969) have found nearly the same value (0’r-015);but the order is in
agreement with the one provided by theoretical models (Jeffreys 1959).
We also found by fitting an expression of the form (21) to the group-differences an
error of the order of 0“.02 in the annual nutation correction, that is bigger than the
one provided by theory (Melchior 1973), and an error in the semi-annual nutation
correction of the order of 0”-015, somewhat smaller than the O”aO25 found by
Popov (1959).
Other similar analyses of a very long homogeneous series of observations would
be necessary to confirm our results. We have principally examined here a suitable
method of analysis for this kind of search and have shown that errors in the main
nutation corrections can thus be revealed.
2. Nearly-diurnal wobble
We have shown in the preceding Sections 1 and 3 that a nearly-diurnal wobble of
amplitude yio and period Po,io, appears in latitude and time observations, made at a
constant mean UT, first, by the ‘ direct effect ’ of amplitude yio and period PI, io
and secondly, .by the ‘ indirect effect ’ of amplitude Ai, and period P 3 , io, yio being
negligible in comparison with Ai,.
Thomas (1964), DCbarbat (1969, 1971), Sugawa & Ooe (1970) with harmonic
analysis, have found peaks which they considered to be the direct effect of the nearlydiurnal wobble. But, if correct, there should exist a further term, with an amplitude
Ri, = kio/(l+ki,) times larger, at the P 3 , io period due to the indirect effect. These
authors find that yi, is of the order of 0‘.01 and PI, io is either - 170 d, or -204 d;
then, from Table 1 we can conclude that Rio N 400 and that the corresponding
periods P3, io are respectively -2500 d and - 1700d. Though these periods are very
long, it is probable that a term of amplitude 4” should have been revealed in the
observations. Therefore it seems unlikely that these peaks are due to the direct effect
of the nearly-diurnal wobble. Maybe these peaks are due to the indirect effect of the
nearly-diurnal wobble; this would change the period Po,io to -23.82 sidereal hours
(sh) and the amplitude yio would be 0”-01/400.
587
Nearly-diurnal wobble of the Earth’s axis
However, in the case of latitude and time analysis of the Paris astrolabe observations (Dtbarbat 1971; Chollet & Dtbarbat 1972) the peaks found by the authors at
170d in latitude and 203d in time (which have been found again in the spectral
analyses of this work) have not given the indirect terms of periods PIz, i o :
PIz,
io =
320d if P 3 , io =
P Z ,io = 463d if P3, io
+ 170d,
= +203d,
PtZ,io = - 116d if P 3 , io
=
- 170d
P 2 ,io = - 130d if P3, io = -203 d
which suggests strongiy that these peaks are not due to the nearly-diurnal wobble.
We have seen in Section 3 that in the case of observations made at a constant
sidereal time, the direct and the indirect effect of the nearly-diurnal wobble appear at
the same period P z , io. Now, Popov analysed observations of two bright zenith stars
of the same declination (Popov 1963) and found in the observed latitude differences a
term of amplitude 0”.016 and period 463 d. He deduced that this term was due to
the direct effect of the nearly-diurnal wobble and that yio = (i”.016 for P 2 , = 463 d.
But as we have shown that the indirect effect must appear at the same period P 2 , io,
may be that in fact Aio = 0”.016 and hence yio = 0”-016/463, the period Po, io
deduced for the nearly-diurnal wobble being unchanged.
Moreover, the results obtained by our analysis show that there do not exist terms
of the form (12) with amplitudes Aiobigger than 0”.015; they would give an amplitude
yio smaller than
and therefore 100 times lower than the one found by all these
authors.
Our results do not permit any conclusion on the period of the nearly-diurnal
wobble. We can only note the period P 3 , near 108 d which would correspond to
the P z , period near 155d, these errors appearing respectively in the four spectral
analyses and in the least-squares fit of expressions (21). They would correspond to a
period Po, of -24.16sh near the -24.164sh period given by the third model of
Jeffreys & Vicente (1964).
Departement d’dstronomie Fondamentale,
Observatoire de Paris-Meudon,
92 190-Meudon,
France.
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