# Pole movement

Pole movement of the North Pole from 1909 to 2001; the positive x-axis (upwards) corresponds to the date line, the positive y-axis (to the right) the 90th degree of longitude east.

As polar motion or Polschwankung (engl. Polar Motion ) denote astronomers and geoscientists a slow swinging displacement of the axis of the earth within the earth's body. Although it only makes up a few millionths of the Earth's radius , it is of great importance for today's geosciences - especially geodesy and geophysics - as well as for astronomy and its fundamental system .

Some astronomers already suspected 150 years ago that the north and south poles of the earth are not completely immutable. It is now known that the North Pole moves in an almost annual, spiral-shaped oscillation of a few meters in amplitude over the earth's surface and also drifts around 10–12 meters per century towards 80 ° West. The latter is also known as polar migration .

The periodic slight tumbling (colloquially "eggs") of the earth's axis is due to the fact that the axis of rotation and the main axis of inertia do not quite coincide and the earth's body is somewhat elastic . Therefore, the earth's body reacts slightly to seasonal or tectonic mass shifts. The polar movement must be taken into account for precise coordinate and reference systems of the earth and space (see ICRS and ITRF ).

The point of intersection of the axis of rotation with the earth's surface, averaged from 1900 to 1905, has been defined as the reference point for the geographic north and south poles and is called CIO (Conventional International Origin). The pole movement is not to be confused with the changing orientation of the axis of rotation in space (see celestial pole ). The gravitational forces of the moon, the sun and the other planets exert torques on the earth's body that lead to a precessional movement of the earth's axis in 26,000 years. In contrast, the magnetic poles of the earth are not directly related to the position of the earth's axis; their relocation is much faster.

## Basics

The axis of rotation of the earth has - like the axis of rotation of every stable top - the tendency to maintain its direction in space exactly (see also inertial system ). During the annual orbit around the sun, it always points in almost the same direction - currently to the so-called polar star in the constellation Little Bear - although its annual orbit is inclined to it.

If the earth were completely rigid and not exposed to any external forces, its axis of rotation would point in exactly the same direction for millions of years due to the conservation of angular momentum. In fact, however, the forces of attraction in the planetary system (especially the gravitation of the moon and sun) cause a small, regular tilt, because the earth deviates somewhat from the spherical shape . This so-called precession and a related effect, nutation , are not directly related to pole movement, but are often confused with it.

1. is not a rigid body, but has a certain elasticity ,
2. is not a ball, but flattened by 0.3 percent,
3. seasonal effects and small deformations occur on their surface , and
4. slow mass shifts take place in the interior of the earth .

The inner and outer changes in shape of the earth (factors 3 and 4) mean that the axis of the greatest moment of inertia also changes with the mass distribution : the earth's axis reacts with a slight wobble , so that the geographical latitudes and lengths of measuring stations can no longer be considered unchangeable . The previously clearly defined relationship between the earth's fixed and the celestial coordinate system (extended earth axis = celestial pole ) becomes more complicated if one wants to take today's demands and high measurement accuracy into account.

## Historical

Although the polar movement is only a few meters, the effect was observed by Friedrich Wilhelm Bessel as early as 1844 , suspected to be the cause around 1860 and proven in 1885 by the Bonn astronomer Karl Friedrich Küstner through precise measurements of the "pole height" ( astronomical latitude ). The assumed changes in the Earth's axis could be filtered out from longer series of measurements as small periodic changes in latitude of around ± 0.3 ″. It was not until the end of the 19th century that astro-geodetic measurements with passage instruments and zenith telescopes had reached this level of accuracy.

## Periods and interpretation of the pole movement

The distance covered by the earth's instantaneous axis of rotation in the years 2001 to 2005 at the North Pole

In addition to Küstner, the most precise empirical studies of the polar movement were carried out by the Viennese geodesist Richard Schumann and the US astronomer Seth Carlo Chandler . The latter discovered the Chandler period in 1891 , which is about 435 days. In contrast, Leonhard Euler had derived a theoretical value of around 305 days for a rigid earth (Euler period). The theoretical basis for this is the following:

The earth's axis of symmetry does not exactly coincide with its axis of rotation , which runs through the center of gravity of the earth's body. That is why the earth's body “wobbles” a little with respect to its own axis of rotation, which is noticeable in changes in the geographic coordinates of a stationary observer. Such a tumbling is only stable if the rotation (approximately) about the body axis takes place with the greatest or the smallest moment of inertia. If this is not the case, the tumbling increases in the long term and the rotating body orients itself around until one of the two body axes mentioned (approximately) coincides with the axis of rotation. Since the axis of symmetry of the earth is the axis with the greatest moment of inertia due to the flattening of the earth , such instability does not occur. The deviation between the axis of symmetry and the axis of rotation therefore remains limited and the axis of symmetry performs a precession-like movement around the axis of rotation about once a year.

The behavior of the “earth spinning top ” and its slight tumbling can be calculated precisely with the methods of scientific mechanics if one assumes a rigid solid body with the dimensions of the earth for it . If, on the other hand, the earth's body (especially the plastic interior ) gives in a little - which geology has long known based on folded rocks - the period of this tumbling is extended because the disruptive forces can no longer be so "grippy".

The total oscillation is made up of a free and a forced component. The free oscillation has an amplitude of approx. 6 m and a period of 415 to 433 days (Chandler period). The fluctuation of the period duration is related to seasonal effects ( leaf fall and vegetation, icing , plate tectonics , etc.), which can be traced back to mass shifts on the earth's surface or in the interior of the earth. The rigidity of the earth's body can be calculated from the difference between the Chandler and Euler periods , but this is made considerably more difficult by their stratification in the earth's crust and mantle . However, the deformability of the earth can also be determined using other methods, e.g. B. by means of the earth tides .

The forced oscillation has an amplitude about half as large and an annual period. It is stimulated by seasonal shifts in water and air masses. The superposition of the two oscillations of different duration means that the amplitude of the overall oscillation fluctuates between approx. 2 m and approx. 8 m every six years.

This spiral movement is superimposed by smaller oscillations with periods of a few hours up to decades . Small spontaneous shifts of a few centimeters can also sometimes be detected - triggered for example by the seaquake of December 26, 2004 near Sumatra , which triggered the massive tsunami in the Ind.

The center of the oscillation drifts at a speed of about 10 m per century in the direction of 80 ° west. This movement is attributed to large-scale tectonic processes.

## International Broad Service, IPMS and IERS

In order to examine these effects more closely, the International Broad Service was founded in 1899 . It consisted of five observatories on different continents , all of which were located at 39.8 ° north latitude. By measuring the astronomical latitude every evening, a continuous curve of the pole movement was obtained, whereby the small (inevitable) contradictions in the data of opposite continents were minimized by means of an adjustment calculation.

A few years after the beginning of space travel , astronomical measurements could be supplemented by methods of satellite geodesy , and soon also improved. The International Polar Motion Service was founded for this purpose (abbreviation IPMS). In the 1990s he switched to the Earth Rotation Service IERS , the results of which are now based on data from five to six very different measurement methods.

See special article: Fundamental Astronomy .

## theory

### Seasonal wave

Figure 3. Vector m of the seasonal component of the polar motion as a function of the season. The numbers and the line marks indicate the beginning of a calendar month. The dashed line is the direction of the major axis of the ellipse. The line in the direction of the minor axis is the position of the excitation function as a function of the season. (100 mas (milliarc seconds) = 3.09 m on the earth's surface).

Today there is general agreement that the seasonal component is a forced oscillation, which is mainly due to atmospheric dynamics. Responsible for this is a standing, antisymmetric atmospheric tidal wave of the meridional wave number and the period of one year, which has a pressure amplitude in a first approximation proportional to (with the pole distance ). In winter in the northern hemisphere there is a high pressure area over the North Atlantic and a low pressure area over Siberia with temperature differences of up to 50 °. In summer it's the other way around. This means a mass imbalance (imbalance) on the earth's surface, which results in a gyratory movement of the figure axis of the earth's body against its axis of rotation. ${\ displaystyle m = 1}$${\ displaystyle \ sin \ theta \ cos \ theta}$${\ displaystyle \ theta}$

The position of the pole movement can be found from Euler's equations . The position of the vector m of the seasonal component of the polar movement describes an ellipse (Fig. 3), whose major and minor axes are in relation

${\ displaystyle {\ frac {m_ {1}} {m_ {2}}} = \ nu _ {C} \,}$
(1)

stand (with the Chandler resonance frequency; or a Chandler resonance period of sidereal days = 1.20 sidereal years). The result in Fig. 3 is in good agreement with the observations. The pressure amplitude of the atmospheric wave that produces this wobble is with a maximum at = −170 ° longitude. ${\ displaystyle \ nu _ {C} = 0 {,} 83}$${\ displaystyle \ tau _ {C} = 441}$${\ displaystyle p_ {o} = 2 {,} 2 \, \ mathrm {hPa}}$${\ displaystyle \ lambda _ {o}}$

It is difficult to determine the influence of the ocean. Its effect has been estimated at 5–10%.

### Chandler wobble

While the annual component remains fairly constant from year to year, the observed Chandler period fluctuates significantly over the years. This fluctuation is determined by the empirical formula

${\ displaystyle m = {\ frac {3 {,} 7} {(\ nu -0 {,} 816)}}}$(for ) ${\ displaystyle 0 {,} 83 <\ nu <0 {,} 9}$
(2)

pretty well described. The amount m increases with the reciprocal frequency. One explanation for the Chandler wobble is the excitation by quasi-periodic atmospheric dynamics. In fact, a quasi-14-month period has been extracted from a coupled atmosphere-ocean model, and a regional 14-month signal has been observed in the ocean surface temperature.

For the theoretical treatment of Euler's equations, the (normalized) frequency must be replaced by a complex frequency , the imaginary term simulating dissipation effects due to the elastic body of the earth. As in Fig. 3, the solution consists of a prograde and a retrograde circularly polarized wave. The retrograde wave can be neglected for frequencies , and the circularly polarized prograde wave remains, with the vector m moving on a circle counterclockwise. The calculated amount of m is ${\ displaystyle \ nu}$${\ displaystyle \ nu + i \ nu _ {D}}$${\ displaystyle \ nu _ {D}}$${\ displaystyle \ nu <0 {,} 9}$

${\ displaystyle m = 14 {,} 5 \, p_ {o} \, {\ frac {\ nu _ {C}} {\ sqrt {{{((\ nu - \ nu _ {C})} ^ {2 } + \ nu _ {D} ^ {2})}}} \,}$(for )${\ displaystyle \ nu <0 {,} 9}$
(3)

This is a resonance curve whose edges go through

${\ displaystyle m \ approx 14 {,} 5 \, p_ {o} \, {\ frac {\ nu _ {C}} {| \ nu - \ nu _ {C} |}}}$(for )${\ displaystyle {(\ nu - \ nu _ {C})} ^ {2} \ gg \ nu _ {D} ^ {2} \, \,}$
(4)

can be approximated. The maximum amplitude of m at becomes ${\ displaystyle \ nu = \ nu _ {C}}$

${\ displaystyle m _ {\ mathrm {max}} = 14 {,} 5 \, p_ {o} \, {\ frac {\ nu _ {C}} {\ nu _ {D}}} \,}$
(7)

In the area of ​​validity of the empirical formula Eq. (2) there is good agreement with Eq. (4). The observations of the past 100 years show that no value greater than mas (milli-seconds) has been found so far . That gives a lower limit for years. The corresponding pressure amplitude of the atmospheric wave is . This number is indeed small and indicates the resonance effect around the Chandlerian resonance frequency. ${\ displaystyle m _ {\ mathrm {max}} \ sim 230 \,}$${\ displaystyle \ tau _ {D} = {\ frac {1} {\ nu _ {D}}} \ geq 100 \,}$${\ displaystyle p_ {o} \ sim 0 {,} 2 \, \ mathrm {hPa}}$

## Pole coordinates

The Earth's rotation parameters, which are regularly determined by the IERS, describe the current rotation speed of the earth and the current orientation of the rotation axis in space as well as the current orientation of the earth's body with respect to the rotation axis.

The polar coordinates and indicate the position of the current axis of rotation (more precisely: the Celestial Ephemeris Pole) with respect to a certain fixed point on the earth's surface (the IERS reference pole) . The x-axis runs in the direction of the prime meridian (more precisely: the IERS reference meridian ) and the y-axis in the direction of 90 ° west. Milli-arcseconds are usually used as the unit of measurement (the distance between the two points on the earth's surface can also be expressed in meters). ${\ displaystyle x}$${\ displaystyle y}$

In the course of the pole movement, the position of a stationary observer also changes with respect to the axis of rotation. If this observer carries out an astronomical determination of his geographical coordinates, he receives slightly different results depending on the current position of the axis of rotation. If its mean astronomical longitude and its mean latitude and the instantaneous axis of rotation has the polar coordinates and , then the ascertained deviations from its mean coordinates are a first approximation ${\ displaystyle \ lambda _ {m}}$${\ displaystyle \ varphi _ {m}}$${\ displaystyle x}$${\ displaystyle y}$

${\ displaystyle \ Delta \ varphi = x \ cos \ lambda _ {m} -y \ sin \ lambda _ {m}}$
${\ displaystyle \ Delta \ lambda = (x \ sin \ lambda _ {m} + y \ cos \ lambda _ {m}) \ tan \ varphi _ {m}}$.

## Geophysical Implications

If one imagines the earth as an exact, rigid sphere and rotating completely symmetrically , the earth's axis would be unchangeable and the north and south poles would remain in the same position on the sphere. In fact, the earth has

As a result, the true axis of the earth makes very complicated, but mostly periodic movements. The mass displacement on and in the earth and the resilience of the earth's body can be calculated partly theoretically, partly empirically by means of suitable physical models and refined a little every year.

In 2003, the Descartes Prize was awarded to Prof. Véronique Dehant by the Observatoire royal de Belgique and a Europe-wide group of around 30 researchers who, under their leadership, developed an expanded theory of Earth's rotation and nutation.

## Applications and reductions

In addition to research , precise knowledge of the position of the earth's axis is also required for several practical purposes. These include satellite navigation , geoid determination , the reduction of geodetic precision measurements (see vertical deviation ) and space travel . If the current pole position were not taken into account, errors in the position of more than 10 meters would result. For example, a 100 km large would power a land surveying suffer differences from cm to dm, or a Mars rocket would miss their target by several 1,000 kilometers.

## literature

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2. a b H. Volland: Atmosphere and Earth's Rotation. Surv. Geophys., 17 , 101, 1996
3. K. Lambeck: The Earth's Variable Rotation: Geophysical Causes and Consequences. Cambridge University Press, Cambridge 1980
4. ^ A b H. Jochmann: The Earth rotation as a cyclic process and as an indicator within the Earth's interior. Z. geol. Wiss., 12 , 197, 1984
5. JM True: The effects of the atmosphere and oceans on the Earth's wobble - I. Theory. Geophys. Res. JR Astr. Soc., 70 , 349, 1982
6. ^ S. Hameed, RG Currie: Simulation of the 14-month Chandler wobble in a global climatic model. Geophys. Res. Lett., 16 , 247, 1989
7. ^ I. Kikuchi, I. Naito: Sea surface temperature analysis near the Chandler period. Proceedings of the International Latitude Observatory of Mizusawa, 21 K , 64, 1982