An instability of the trigonometric solution for the periodical components of the polar motion
Estimation of the real errors of the polar motion approximation with trigonometric functions is made by a model of simulated polar motion with variable amplitudes of the Chandler and seasonal components. It is shown that the real errors of the trigonometric coefficients are much more than the formal...
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| Published in: | Кинематика и физика небесных тел |
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| Date: | 2005 |
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Головна астрономічна обсерваторія НАН України
2005
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| Cite this: | An instability of the trigonometric solution for the periodical components of the polar motion / Ya. Chapanov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 351-354. — Бібліогр.: 3 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860114372861362176 |
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| author | Chapanov, Ya. |
| author_facet | Chapanov, Ya. |
| citation_txt | An instability of the trigonometric solution for the periodical components of the polar motion / Ya. Chapanov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 351-354. — Бібліогр.: 3 назв. — англ. |
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| container_title | Кинематика и физика небесных тел |
| description | Estimation of the real errors of the polar motion approximation with trigonometric functions is made by a model of simulated polar motion with variable amplitudes of the Chandler and seasonal components. It is shown that the real errors of the trigonometric coefficients are much more than the formal estimation errors, obtained by the least-squares method (LSM).
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| first_indexed | 2025-12-07T17:35:21Z |
| format | Article |
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AN INSTABILITY OF THE TRIGONOMETRIC SOLUTION
FOR THE PERIODICAL COMPONENTS OF THE POLAR MOTION
Ya. Chapanov
Central Laboratory for Geodesy, BAS
Acad. G. Bonchev Str., bl.1, 1113 Sofia, Bulgaria
e-mail: astro@bas.bg
Estimation of the real errors of the polar motion approximation with trigonometric functions is
made by a model of simulated polar motion with variable amplitudes of the Chandler and seasonal
components. It is shown that the real errors of the trigonometric coefficients are much more than
the formal estimation errors, obtained by the least-squares method (LSM).
INTRODUCTION
The determination of the Earth’s orientation parameters (EOP) is important part of the modern scientific
investigations. The first determinations of the EOP only include the pole coordinates from optical astrometry,
and since 1956 those of the pole coordinates, the Universal Time and the celestial pole offsets. The accuracy of
the pole coordinates determination from the optical astrometry varies within 0.01 and 0.03 arcsec. In the last
two decades the optical observations are replaced by various space and ground-based observational techniques,
which improves significantly the estimation accuracy. The accuracy of the C04 solution of the IERS for pole
coordinates since 1996 is 0.2 mas.
Usually, the analysis of the polar motion consists of separation and investigation of the two main periodical
components namely seasonal and Chandler oscillations. The successful separation of the seasonal and Chandler
oscillation is possible over six-year or longer time spans of observations, due to almost resonance relation
5 : 6 between the annual and Chandler frequencies. The most used mathematical model for determination of
the periodical components of the polar motion, involved by [1] and [2], consists of trigonometric functions of
common type
x = x0 +
n∑
i=1
aai sin iωat + bai cos iωat +
m∑
i=1
aci sin iωct + bci cos iωct, (1)
y = y0 +
n∑
i=1
cai sin iωat + dai cos iωat +
m∑
i=1
cci sin iωct + dci cos iωct, (2)
where x and y are the pole coordinates, x0 and y0 are the mean coordinates in the middle of the six-year time
interval, aai, bai, cai, dai, i = 1, . . . , n; acj, bcj, ccj, dcj , j = 1, . . . , m are unknown harmonic coefficients of
the components with known seasonal (annual) frequency wa and Chandler frequency wc; t is the observation
epoch. Usually, the number of harmonics is equal to 1 (n =m =1), for study of the semi-annual and semi-
Chandler oscillations the value of harmonics n and m may increase to 2.
SEASONAL AND CHANDLER AMPLITUDE VARIATIONS, DETERMINED
FROM THE SOLUTION OA04 FOR THE POLE COORDINATES
The recent Vondrak and Ron solution OA04 for the pole coordinates [3] is obtained from all optical observations
for the period 1899.7–1992.0 using the combined star catalogues, based on Hipparcos catalogues and ground
observations (Fig. 1). The variations of the amplitudes of the seasonal and Chandler oscillations of the polar
motion are determined from the solution OA04 by the model (1, 2). The estimates are obtained by the LSM with
running 6-year interval. The variations of the amplitude of the seasonal oscillations of the polar motion consist
of decadal oscillation within the limits 0.05′′–0.15′′ (Fig. 2). The variations of the amplitude of the Chandler
oscillations are from 0.0′′ to 0.25′′ with periods between 20 and 40 years (Fig. 3).
c© Ya. Chapanov, 2004
351
Figure 1. The last solution OA04 of Vondrak and Ron [3] for the pole coordinates X, Y and their errors Mx, My
Figure 2. Variations of the amplitude of the seasonal oscillations of the polar motion (the graph below, solid line for
x-coordinate, dashed line for y-coordinate), determined by the LSM estimation, according to the model (1, 2). The upper
graph represents the formal estimation errors
Figure 3. Variations of the amplitude of the Chandler oscillations of the polar motion (the graph below, solid line for
x-coordinate, dashed line for y-coordinate), determined by the LSM estimation, according to the model (1, 2). The upper
graph represents the formal estimation errors
352
SIMULATION OF THE POLAR MOTION WITH VARIABLE SEASONAL
AND CHANDLER AMPLITUDES
To estimate the real errors of the approximation of the polar motion with trigonometric functions of the type (1, 2)
it is very useful to use simulation of the polar motion with known variations of the amplitudes of the seasonal
and Chandler oscillation, which are close enough to its real changes in the time. A simple model of the polar
motion is presented by the equations (3, 4).
x = Aan sin 2πt + Ach cos
2πt
Tch
, y = Aan cos 2πt − Ach sin
2πt
Tch
, (3)
Aan = 0.1′′ + 0.04′′ sin
2πt
10
, Ach = 0.2′′ + 0.075′′ sin
2πt
30
, Tch = 1.18a. (4)
The model (3, 4) includes constant part 0.1′′ of the seasonal amplitude Aan and variations with period
10 years and amplitude 0.04′′. The Chandler amplitude Ach consists of constant part 0.2′′ plus variations with
period 30 years and amplitude 0.075′′. The period of the seasonal oscillation is exactly one year, and the period
of the Chandler oscillation Tch is 1.18a. The simulated polar motion is similar to the behaviour of the real
polar motion for the period 1940–1970 (Fig. 4). The changes of the amplitudes of the seasonal and Chandler
oscillations, according to the model (3, 4) is shown in Fig. 5 by the solid lines: the upper line represents
the Chandler amplitude variations, and the lower line is the seasonal amplitude variations.
Figure 4. Simulation of the polar motion with variable amplitudes of the Chandler and seasonal oscillation (solid line
for x-coordinate, dashed line for y-coordinate)
Figure 5. Simulated variations of the amplitudes of the Chandler (upper solid line) and seasonal oscillation (lower solid
line), and corresponding estimated variations of the amplitudes (with dashed lines), obtained by the model (1, 2)
The amplitudes of the seasonal and Chandler oscillations, which approximate the simulated polar mo-
tion (3, 4) by the model (1, 2), are determined from the estimated trigonometric coefficients by the formulae (5).
Aan,x =
√
a2
ai + b2
ai, Aan,y =
√
c2
ai + d2
ai, Ach,x =
√
a2
ci + b2
ci, Ach,y =
√
c2
ci + d2
ci. (5)
353
Figure 6. Real errors of the estimated amplitudes of the seasonal and Chandler oscillations of the simulated polar
motion. The formal estimation errors are about 10 times lower
The computed seasonal and Chandler amplitudes by equations (5) are shown in Fig. 5 with dashed lines.
It is seen significant differences between simulated variations of the amplitudes and corresponding variations,
obtained from trigonometric approximation by the model (1, 2). These differences are estimated by the real
errors of the polar motion approximations with trigonometric functions. They are shown in Fig. 6 together
with the formal estimation errors, determined by the least-squares method. The real errors of the trigonometric
approximation of the polar motion by the model (1, 2) are almost 10 times higher than the formal estimation
errors. The mean values of the obtained real errors are 0.019′′ for the seasonal amplitude and 0.013′′ for
the Chandler amplitude. The maximum values are 0.036′′ and 0.026′′. An additional computation with shifted
period of the Chandler oscillation with 1/100 is made, and in this case the maximum errors increase up to
0.048′′ for the annual amplitude and 0.028′′ for the Chandler amplitude.
CONCLUSIONS
1. The trigonometric solution for the periodical components of the polar motion is unstable in the case of
time variations of the annual and Chandler amplitudes.
2. This instability of the trigonometric solution increases the maximum value of the real errors of the annual
and Chandler amplitudes up to 48 mas and 28 mas when the Chandler period is shifted with 1/100 and
up to 36 mas and 26 mas with exact knowledge of the Chandler period value.
3. The real errors of the trigonometric solution for the periodical components of the polar motion exceed
about 10 times the formal estimation errors of the annual and Chandler amplitudes.
Acknowledgements. The work was supported by the National Council for Scientific Research at Ministry of
Education and Science of Republic Bulgaria with the grant NZ-1213/02, also by the Organizing Committee of
the MAO-2004 Conference “Astronomy in Ukraine – Past, Present and Future”, Kiev, Ukraine, July 15–17, 2004.
[1] Hattori T. A new study of latitude variation // Japanese J. Astron. Geophys.–1947.–21.
[2] Sekiguchi N. On a character about the secular motion of the mean pole of the Earth // Publs Astron. Soc.
Jap.–1954.–5.–P. 109–113.
[3] Vondrak J., Ron C. The great Chandler wobble change in 1923–1940 re-visited // Proc. Chandler Wobble Workshop,
Luxemburg, April 2004.–2004.
354
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| id | nasplib_isofts_kiev_ua-123456789-79673 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0233-7665 |
| language | English |
| last_indexed | 2025-12-07T17:35:21Z |
| publishDate | 2005 |
| publisher | Головна астрономічна обсерваторія НАН України |
| record_format | dspace |
| spelling | Chapanov, Ya. 2015-04-03T18:29:31Z 2015-04-03T18:29:31Z 2005 An instability of the trigonometric solution for the periodical components of the polar motion / Ya. Chapanov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 351-354. — Бібліогр.: 3 назв. — англ. 0233-7665 https://nasplib.isofts.kiev.ua/handle/123456789/79673 Estimation of the real errors of the polar motion approximation with trigonometric functions is made by a model of simulated polar motion with variable amplitudes of the Chandler and seasonal components. It is shown that the real errors of the trigonometric coefficients are much more than the formal estimation errors, obtained by the least-squares method (LSM). The work was supported by the National Council for Scientific Research at Ministry of Education and Science of Republic Bulgaria with the grant NZ-1213/02, also by the Organizing Committee of the MAO-2004 Conference “Astronomy in Ukraine – Past, Present and Future”, Kiev, Ukraine, July 15–17, 2004. en Головна астрономічна обсерваторія НАН України Кинематика и физика небесных тел MS4: Positional Astronomy and Global Geodynamics An instability of the trigonometric solution for the periodical components of the polar motion Article published earlier |
| spellingShingle | An instability of the trigonometric solution for the periodical components of the polar motion Chapanov, Ya. MS4: Positional Astronomy and Global Geodynamics |
| title | An instability of the trigonometric solution for the periodical components of the polar motion |
| title_full | An instability of the trigonometric solution for the periodical components of the polar motion |
| title_fullStr | An instability of the trigonometric solution for the periodical components of the polar motion |
| title_full_unstemmed | An instability of the trigonometric solution for the periodical components of the polar motion |
| title_short | An instability of the trigonometric solution for the periodical components of the polar motion |
| title_sort | instability of the trigonometric solution for the periodical components of the polar motion |
| topic | MS4: Positional Astronomy and Global Geodynamics |
| topic_facet | MS4: Positional Astronomy and Global Geodynamics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79673 |
| work_keys_str_mv | AT chapanovya aninstabilityofthetrigonometricsolutionfortheperiodicalcomponentsofthepolarmotion AT chapanovya instabilityofthetrigonometricsolutionfortheperiodicalcomponentsofthepolarmotion |