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Lezioni Terza Settimana

Esercizi in aula (non riportati qui) I movimenti

dell'osservatore terrestre Precessione degli

equinozi Nutazione Nutazione libera o euleriana

(il moto del Polo) Questa parte è quasi tutta

scritta in inglese!

Tempo e anno siderale

Premettiamo due concetti che approfondiremo in

seguito Giorno siderale intervallo di tempo

tra due consecutivi passaggi del punto ? in

meridiano. E' lievissimamente più corto del

periodo di rotazione della Terra (cioè del

passaggio in meridiano di una stella equatoriale

priva di moto proprio), ma per ora possiamo

confondere i due periodi. E' invece nettamente

diverso dall'intervallo di tempo tra due passaggi

del Sole in meridiano, che è più lungo in media

di circa 3m56s. Anno tropico intervallo di

tempo tra due consecutivi passaggi del Sole per

il punto ?. E' più corto del vero periodo di

rivoluzione della Terra attorno al Sole causa la

precessione dell'equinozio che ora discutiamo.

The movements of the terrestrial observer

The equatorial and ecliptic coordinates are based

respectively on the plane of the celestial

equator and of the ecliptic, having a common

origin in the vernal point ? every movement of

these planes with respect to the fixed stars will

result in a time variation of the coordinates.

The stars ideally provide a fixed reference

frame among which we will determine the complex

movements of the observer. For fundamental

dynamical reasons, the ecliptic plane is much

more stable than the equator, whose movements are

larger, and known with some residual imprecision

even today. Those minute uncertainties, however

small, are of great interest for the astronomer,

because they somewhat hamper the precise

knowledge of the system of motions and of the

overall field of forces of the Milky Way. Hence

the efforts, not only by geophysicists but also

by astronomers, to know with even better

precision the movements of the terrestrial

observer.

First dynamical considerations - 1

The equatorial system (?, ?) is the one used in

high precision positional catalogues. However it

depends on the orientation in space of the Earth

and its rotation (position of the equatorial

plane with respect to the fixed stars, meridian,

Sidereal Time), and on the revolution around the

Sun (ecliptic, point ?). The motion of the

Earth can be considered as a combination of two

unrelated motions, namely a translation of the

center of mass and a rotation of the figure

around an axis passing through the barycenter.

First dynamical considerations - 2

In first approximation, the barycenter of the

Earth revolves around the Sun as a point-like

particle subject to the gravitational pull of the

Sun and of the other planets. With better

precision, it is the Earth-Moon (E-M) barycenter

that follows Keplers laws (it may be worth

recalling that even in the absence of the other

planets, the EM barycenter would follow an orbit

very slightly different from an ellipse).

Therefore, it is preferable to identify the

ecliptic with the orbital plane of this E-M

barycenter, and free it from the periodic

perturbations due mostly to Venus and Jupiter.

The Sun is never more than 2 above or below this

mean ecliptic, a fact that justifies the simpler

treatment of the previous Chapters.

Unfortunately, different authors have slightly

different definitions of ecliptic.

Orientation and rotation of the Earth - 1

Several factors complicate the dynamical problem

- the instantaneous rotation axis is influenced

by the presence of the Moon and of the Sun

(precession and nutation), - the instantaneous

rotation axis doesnt necessarily coincide with

the minor axis of the geometrically best-fitting

ellipsoid (free or Eulerian nutation). - the

distribution of the masses is not spherical nor

has an azimuthal symmetry, nor is absolutely

constant in time. Due to the first factor, the

instantaneous rotation axis passes through the

barycenter, but its orientation varies with

respect to the fixed stars, causing a variation

of their equatorial coordinates (precession and

nutation). The variation of the position of

the rotation axis with respect to the axis of the

ellipsoid (free or Eulerian nutation) causes a

small variation of the astronomical latitudes of

each Observatory, and a wandering of the

astronomical poles around the geodetic poles

confined to a circle of some 18 m diameter.

Orientation and rotation of the Earth - 2

The mass of the whole Earth is M? 5.976x1027

g, the mass of the ocean is approximately 10-4 M?

, and that of the atmosphere about 10-7 M? .

Therefore a non-negligible fraction of the

Earth's surface is non-solid. Furthermore,

even the solid Earth is not perfectly rigid, and

the distribution of masses can change both at the

surface (winds, tides, currents) and in the

interior (earthquakes). While the meteorological

effects have small and periodic effects,

occasional movements of the internal masses can

cause abrupt changes of the rotation with respect

to the figure. The rotation cannot therefore be

uniform for arbitrarily long periods, at the

present epoch we witness a secular decrease of

the angular velocity, with over-imposed periodic

fluctuations but also with abrupt changes. Even

the overall figure of the earth at the present

time is changing, the Earth becomes more

spherical than only a decade ago. To be sure, the

effects of these internal factors are small, even

negligible for many astronomical applications,

but they are well measurable with todays

instruments.

The precession of the equinox - 1

The dominant variation of stellar coordinates

was discovered in 129 B.C. by Hipparchus,

comparing his own determination of the ecliptic

coordinates of Spica (? Vir) with those derived

144 years earlier by Timocharis while the

ecliptic latitude had remained constant, the

longitude had increased by some 2 (namely by

approximately 50.4 per year). The same variation

was soon found on all stars. Hipparchus put

forward the following explanation for the

increase of longitudes the whole sphere of the

fixed stars has a rotation in direct

(anti-clockwise) sense around the ecliptic pole.

The Sun would therefore encounter ? each year

somewhat earlier in the words of Hipparchus, the

ingress of the Sun in ? would precede each year

by the time taken to describe an arc of 50.4.

Hence the expression precession of the equinox.

Through the centuries, the constellation where

the Sun is seen to ingress in ? changes it was

Aries at the time of Hipparchus, it is Pisces

today, another 2000 years and it will be

Acquarius. Some 1600 years passed before

Copernicus gave the right interpretation of

precession, the Earths rotation axis describes a

cone of semi-aperture ? around the ecliptic pole

E, in a period of approximately 25800 years (

360/50.3 y-1, often called a platonic year).

The precession of the equinox - 2

The celestial pole P is therefore seen at each

time on a point of the small circle distant ?

from the ecliptic pole E, as shown in the Figure.

In other words, the parallel of ecliptic latitude

? 90- ? is the locus described by P in 25800

years. The South celestial pole is seen

diametrically opposed on the corresponding circle

having ? -(90- ? ).

Figure Two representations of the precession of

Hipparchus. Left, the vector from the center of

the Earth to the celestial North pole describes a

cone of fixed semi-aperture ? around the

Ecliptic North Pole, in retrograde direction.

Right, the instantaneous motion of the celestial

pole is a vector tangent both to the small circle

distant ? from the ecliptic pole and to the

equinoxial colure.

The position of the celestial poles

During this movement, the celestial poles will

be seen in different constellations today, the

celestial North pole approaches the bright star ?

Umi (Pole star). The present distance of about

45 will decrease to a minimum value of 27 in

2102, and then it will progressively augment. As

already said, there is no correspondingly bright

star near the present position of the celestial

South pole (the one used by the Astronomical

Almanac as pole star for obtaining latitudes is ?

Octanctis, of visual magnitude 5.5).

The dynamical explanation of the precession given

by Newton (1687)

The phenomenon described by Hipparchus and

Copernicus was explained on dynamical basis by

Newton in his Principia (1687) the figure of the

Earth cannot be a perfect sphere (more precisely,

the internal mass distribution cannot have

spherical symmetry), it must be a spheroid,

having equatorial axis larger than the polar one.

Therefore the Sun, which is usually not on the

equatorial plane, exerts a torque on the Earths

rotation axis that makes the axis move

perpendicular to the instantaneous plane passing

through the axis itself and the direction to the

Moon (often, an analogy with a spinning top is

made however, the sense of precession and

rotation are opposite, the spinning top precesses

and rotates in the same sense, the Earth

precesses in the opposite sense of the diurnal

rotation).

Solar precession, winter solstice.

The luni-solar torque

Notice the dependence from the inverse cube of

the distance, from the mass and from the

declination of the Sun. The Moon must exert the

same effect however, because the amplitude of

the torque is proportional to the mass of the

responsible body and to the inverse cube of its

distance form the Earths barycenter, the

magnitude of the lunar effect is slightly more

than twice that of the Sun (in other words, the

Moon is responsible for 2/3, the Sun for 1/3 of

the total). Therefore the collective term

luni-solar precession is usually employed.

The nutation

Copernicuss description, and Newtons

explanation, must be only partially correct, for

two reasons first, the Moons orbit is inclined

by 59 to the ecliptic plane second, the

distance to the Moon and (to lesser extent) that

to the Sun change during the lunar month and the

tropical year. As a consequence, the true motion

of the celestial pole does not rigorously follow

the small circle at a distance ? from the

ecliptic pole a series of cyclic terms of

different amplitudes and periods, affecting also

the instantaneous obliquity, must be present.

These effects however do not accumulate as the

luni-solar precession does, and they went

undetected until the advent of telescopic

observations. The credit for their discovery goes

to the English astronomer J. Bradley, in the

XVIII Century, using a long series of

measurements of the declination of the bright

star ? Dra (not too distant from the ecliptic

pole E) the declination of the star, once

corrected for the luni-solar precession, appeared

to increase by 18 from 1727 to 1736, and to

decrease by the same amount from 1736 to 1745, as

if the celestial pole had an oscillatory movement

(called nutation by Bradley, the same term used

for the oscillation of a ships masts) around a

mean position, of amplitude ? 9 and main period

of 18.6 years. This period is exactly the same of

the retrogradation of the nodes of the lunar

orbit on the ecliptic.

The movements of the ecliptic

Until now, we have assumed that the plane of the

ecliptic is constant with respect to the distant

stars, but this assumption was discovered not to

be true at the level of accuracy reached by XVIII

Century observational astronomy. In the same

years, Euler had predicted the movement of the

ecliptic plane under the influence of the

planets, in particular of Venus and Jupiter.

The obliquity of the ecliptic is seen at the

present epoch to have a secular decrease of about

0.5 per year therefore, from the times of

Copernicus to today, ? has considerably

decreased. Similar to the luni-solar

precession, the influence of the planet is a

periodic effect, but the period is so long (some

110 000 years) that for millennia it will

accumulate, just as the luni-solar precession

does. We are therefore justified to call it

planetary precession, and to add its magnitude

into the description of the general precession

(because of its negative sign, the planetary

precession makes the constant 50.4 become

slightly smaller). A planetary nutation must

also be present however, the movements of the

ecliptic with respect to the equator do not

change the stellar declinations, but only the

common origin of the right ascensions, so that

the planetary nutation goes unnoticed in

differential measurements.

The movements of the fundamental planes - 1

Notice the movement of the equator is much

larger than that of the ecliptic!

- Consider in the Figure the two fundamental

planes (equator and ecliptic) at two dates t1 and

t2 (with t2 later than t1 in order to fix the

sense of the movement) intersecting in ?1 and ?2,

with obliquities ?1 and ?2 respectively. Each

element in the figure, e.g. the angle J, will be

a function of time J J(t), which can be thought

of as composed of two parts - a secular one (actually a periodic one but with

very long period) - a periodic one, with longest period equal to 18.6

years, and many shorter ones.

The movements of the fundamental planes - 2

During several decennia, or even few centuries,

the first (secular) part can be developed in a

time series with only the first few terms of

importance. Therefore, we can write

where the secular terms in t, t2, t3 etc.

(namely, the precession terms) are zero at the

initial epoch, but the short period ones n(t),

n(t0) (namely the nutation terms) are not

necessarily so. The instantaneous elements are

called true elements (true equator, true equinox,

true obliquity, etc.) When we free the elements

by the effects of nutation, namely we consider

the elements affected only by the lunisolar

precession, we obtain the mean elements (mean

equator, mean equinox, mean obliquity, etc.)

First Order effects of precession - 1

We shall call luni-solar precession, without

other adjectives, that due to a constant

luni-solar torque on a rigid Earth, whose effect

would be a strictly periodic rotation of the

celestial pole around the ecliptic one, with

constant velocity and obliquity, namely a

progressive, uniform increase of all the

longitudes by

where ? is the notation favored by many authors.

It is evident however that ? is an angular

velocity (arc per unit time, in this case, the

tropical year). This amount derives from the

dynamical figure of the Earth (moments of

inertia), from the obliquity of the ecliptic and

from the distances, orbital elements and masses

of the forcing bodies. We have already said

that 2/3 of the value of ? derive from the Moon

and 1/3 from the Sun.

First Order effects of precession - 2

In the assumption of a fixed ecliptic, after an

elementary time dt 1 year the moving equator

has performed an elementary rotation dJ around

the diameter MM, and in the same amount of time

the celestial pole has moved from P to PdP along

a great circle perpendicular to the solstitial

colure, and which is also the hour circle of the

initial equinox ?1, so that dP dJ.

The intersection between the ecliptic and the

equator has moved for ?1 to ?2, describing the

elementary precession in longitude

Precession in AR e DEC - 1

The projection of this elementary arc on the

moving equator amounts to the elementary

luni-solar precession in Right Ascension

while the perpendicular component along the Hour

Angle of ?1 is the elementary lunisolar-precession

in Declination

Returning to angular velocities, it is customary

to put

(m 46.21/y 3s.08/y)

(where y is the tropical year)

(n 20.34/y 1s.34/y)

Precession in AR e DEC - 2

Taking the time derivative of the transformation

between ecliptic and equatorial coordinates

and with the initial assumptions that

from the transformation between equatorial and

ecliptic coordinates we easily derive

Precession in RA e DEC - 3

For most applications, one has the task of

precessing the mean coordinates, known at a given

epoch, to the values needed for observing at a

wanted date not too distant from the epoch, say

not more than t 25 years. Let us proceed for

the moment in the pre-1984 way, utilizing a

catalogue (e.g. AGK3 for the northern

hemisphere), that gives the equatorial

coordinates at the mean equinox B1950.0. The task

is to obtain the mean coordinates at epoch

B1950.0t. In a first approximation, the

following formulae will suffice

Notice however that d?/dt becomes very large in

the proximity of the celestial poles, so that the

calculation there becomes critical.

A graphic representation of the first order

precession

In RA

In DEC

The planetary precession

Let us now consider the planetary precession,

namely the effect of the gravitational

perturbations of the planets on the obliquity.

The previous discussion has given us the

possibility to fix in inertial space the

equatorial plane and its pole. Therefore we only

need here to consider the small motion of the

ecliptic pole around the celestial one, which we

can represent with an elementary rotation of the

ecliptic plane about a given point N

This elementary rotation of the ecliptic moves ?

in the direct sense along the equator, by

approximately

that must by subtracted from the value of the

luni-solar precession to obtain the value of the

general precession

As a consequence, the value of constant

slightly decreases

to about 46.07 y-1 3s.07 y-1. The precession

in Declination is not affected.

The diminution of the obliquity ? - 1

There is a second consequence of this rotation of

the ecliptic around a given point N, namely the

slight decrease of the obliquity itself with

reference to the previous figure, consider the

elementary spherical triangle ?N?, where ? is

the position of ? on the fixed equator after one

year, ?A is the known longitude of N

(approximately 174.85) and ?A is the angle in N

(namely the inclination of the mobile ecliptic of

date on the fixed one) from the transformation

laws we get

The diminution of the obliquity ? - 2

are very small angles, so that

After ? years, the inclination of the mobile

ecliptic will be

For instance, at epoch J2001.5 the inclination

was 0.7 with respect to the mean ecliptic of

J2000.0. The obliquity of the ecliptic is

therefore not constant. Tycho Brahe obtained a

reliable determination of ? in 1590, finding the

value ? 2330, while today ? is measured

closer to 2326. As a consequence, none of the

constants ?, m, n, G, g, ?A, ?A, is really a

constant.

Precession in RA and DEC after 1984

Since 1984, the new system of constants adopted

by IAU in 1976 has been enforced in

all Almanacs. Here are some of the new values G

50.290966 0.02222 T , ? 2326'21''.448

- 0''.00468150 T m 46''.124362 0''.02793 T

, n 20''.043109 - 0''.008533 T

where T is the number of Julian centuries of

365.25 days of 86400 seconds starting from the

new fundamental epoch J2000. Furthermore, it has

become customary to calculate the mean

coordinates not for the beginning of the year,

but for its mid-point. Notice that the unit of

time in the new IAU system is not snylonger the

tropical year, but the Julian year. (However,

the ratio between the two years, namely

1.00002136, does not entirely justify the strong

revision of G).

The nutation - 1

We have called nutation the collection of the

short period movements of the equator. The

principal part, discovered by Bradley, is due to

the influence of the Moon, whose orbital plane is

inclined by approximately i 59 to that of the

ecliptic (see Figure we ignore here some smaller

movements of the Moon).

Let us call N the ascending node of this orbit on

the ecliptic, and N that on the equator

(ascending meaning that node where the latitude,

or the declination, passes from negative to

positive values). The two nodes are not fixed in

inertial space N precesses along the ecliptic in

retrograde sense, by approximately 191 each day

(say, 3 lunar diameters per lunation, toward

West, the path of the Moon is very complex one,

and it must be carefully allowed for in studies

such as the lunar occultations) making a full

turn in 18.6 years.

The nutation - 2

Correspondingly, the pole M of the lunar orbit is

seen to describe a small circle of radius i

around the ecliptic pole E, in the same period of

time. As a consequence, the declination of the

Moon varies between approximately ? 18.8 and ?

28.8, according to the longitude of the node

?(N), whose expression is

if t in tropical years since 1900.0, or also

if t is in Julian years after J2000.0. When the

longitude of the node gets to zero (as it did in

1987.8, and it will do in 2006.4), the

declination assumes the minimum or maximum

possible values.

The nutation - 3

Consider now the ascending node N on the

equator, and the spherical triangle PEL N will

oscillate in 18.6 years around ? with an

amplitude of about ?13. Indeed

Therefore, the instantaneous movement of the pole

is no longer along the great circle P?, but along

PN, so that the nutation changes not only the

origin of the longitudes but also the obliquity.

According to Bradleys measurements

The nutation - 4

This movement can be visualized as the

instantaneous pole Pv describing a retrograde

cone around the mean pole Pm, which in its turn

describes a cone of aperture ? around E. Imagine

looking at this movement from the outside of the

celestial sphere, as in the Figure

The ratio of the two axes was justified on

dynamical reasons by dAlembert, who showed that

it must be equal to cos2? /cos?.

On the plane tangent to the celestial sphere in

Pm, the locus occupied by Pv is an ellipse of

semi-major axis ?y 9.2, and of semi-minor axis

?x 17.2 sin? 6.9, described with a period

of 18.6 years in the retrograde sense.

The nutation - 5

The complete phenomenon of nutation contains many

other terms of smaller but non negligible

amplitude the second most important in longitude

has amplitude of 1.32, and in obliquity of

0.57, and indeed these values slightly change

with the epoch. It is customary to indicate the

complete nutation in longitude with the symbol

??, that in obliquity with ??. The nutation in

longitude has the same structure of the

luni-solar precession

The term in obliquity causes a variation of ?,

but it doesnt affect ? after few simple

calculations we obtain

The nutation - 6

and in total

For instance, for the year 2000 and to a

precision of 1, the Astronomical Almanac gives

the following numerical expressions

being d JD - 2451543.5 (2451544 is the JD at

Greenwich noon on Jan. 0, 2000) .

Precession plus nutation

Summing up precession and nutation, after some

manipulation, we obtain

A, B, E being function of the date named

Bessels daily numbers. Still another way of

computing the combined effect of nutation and

precession is by means of the so called

independent day numbers f, g, G. We refer to the

Astronomical Almanac for their expressions.

Newcomb's precessional angles- 1

Let us take as reference a fixed star X, and be

P0 and P two successive positions occupied by the

celestial North pole at time t0 and t. In the

spherical triangle P0PX consider arc ?A and

angles ?A, zA. Angle ?A will be very small for

small (t-t0), and so will be angle zA, because

arc (P0P P?) will differ very little from a

great circle at any rate it will always be ?A ?

zA. Arc ?A is not exactly the path described by

the true pole, which actually is a somewhat

irregular curve. It is

Newcomb's precessional angles- 2

The elements (?A, ?A, zA) are given by

(T in Julian centuries since J2000.0)

The same transformations can be expressed as a

rotation matrix P applied to an initial Cartesian

system (x0, y0, z0) to derive (x, y, z) and

viceversa, namely r Pr0 or r0 P-1r, whose

elements can be found by the above equations for

instance

To allow for nutation, the rotation P will be

followed by rotation RN

Precession and position angles

Precession and nutation are, at least with great

precision, rigid rotations of the celestial

sphere, and as such they do not alter the angular

distance between the stars. Therefore, the

observed shape of a constellation, or of a

nebula, will not be altered (theyll change

because of the proper motions, but this is a

different effect). However, the position angle p

between two objects will change because it is

measured from the variable direction of the North

celestial pole. After some manipulation, it can

be seen that

In turns, the differential coordinates of the two

nearby objects (?1 ? ?2 ? ?, ?1 ? ?2 ? ?) will

change by

The effect is very large in the proximity of the

celestial poles. Therefore, the position angle

must be given in conjunction with an epoch.

Dynamical considerations -1

It is possible to demonstrate that the

gravitational potential due to Sun and Moon, and

responsible for the precession and nutation is

in the usual assumption that the Earth is a

homogeneous spheroid of revolution, with

equatorial and polar inertia moment A and C

respectively, given by

Notice the presence in Uprec of the declination

of the two bodies.

Dynamical considerations -2

Using Kepler's III law (see a later paragraph for

the demonstration), if P is the period and e n

the mean motion (n 2?/P) for each external body

we have

However, the mass of the Sun greatly exceeds that

of the Earth (approximately 330.000 times), while

the mass of the Moon is about 1/80 of that of the

Earth. Therefore, only the mass of the Moon

remains in the total expression

and can be estimated by the precessional

constants.

Comparison between the precession and rotation

potentials

Let us compare now this precessional energy with

that of the diurnal rotation T

Uprec can be maximized by taking the maximum

value of the declination of Sun and Moon, whence

a very modest fraction indeed! We shall examine

the rotation of a free Earth, as if the Moon

and the Sun were not forcing the precession and

nutation.

The rotation of the free Earth - 1

The treatment of this problem, essentially due to

Euler, is very difficult, so that we expound here

only few simple results.

The figure shows a reference system XYZ fixed in

the Earth, and therefore rotating with angular

velocity ? (?1, ?2, ?3) with respect to the

inertial system XoYoZo. The rotational energy is

where ? is the sidereal diurnal velocity. The

plane XY can be identified with the terrestrial

equator, the direction Z with the pole of the

figure, the plane XoYo with the ecliptic, the

direction Zo with the pole of the ecliptic. The

two planes intersect along the lines of nodes

passing through the equinoxes.

Euler's angles

The Eulerian rotation angles. E ecliptic pole, P

celestial pole, N ascending node of the

ecliptic on the equator. ? angle EP angle

X0NX nutation angle 2?-? angle NX0 angle

Y0EP precession angle ? angle NX angle YPB

angle of diurnal rotation.

The relative positions of the rotating frame is

specified by the 3 Euler's angles (?,?,?), whose

derivatives can be expressed as functions of

(?1, ?2, ?3).

The rotation of the free Earth - 2

After several passages we obtain the relations

with ?, q unknown initial constants. On the

other hand, the total angular momentum M

is constant in the inertial space, even if its

direction does not necessarily coincide with that

of ?. However, it can be shown that M, ? and Z

(the direction of the pole of the figure), remain

always in the same plane. Therefore, the rotating

observer, rigidly connected with the Earth, sees

the vectors ? and M in rotation around Z with

constant angular velocity ? while the component

of ? in the equatorial plane has a constant

amplitude ?.

Il moto del polo - 1

Per losservatore terrestre dunque, il polo di

rotazione diurna sembrerà descrivere un moto di

precessione attorno al polo di figura (moto detto

di polodia), con frequenza

pari dunque a circa 10 mesi e ampiezza

determinabile dalle osservazioni, più altri moti

molto minori che qui trascuriamo. Siccome abbiamo

definito la latitudine come altezza del Polo

celeste (quello di rotazione) sullorizzonte, la

conseguenza è una variazione di latitudini con

detto periodo. Tuttavia, le osservazioni provano

che la polodia è composta di due termini, uno con

frequenza annua, e facilmente giustificabile con

la periodica variazione di distribuzione della

massa della Terra (e dunque dei suoi momenti di

inerzia) per cause meteorologiche, e uno con

frequenza di circa 14 mesi, nettamente più lungo

di quello euleriano. La giustificazione di

questa differenza sta nella non perfetta rigidità

della figura terrestre.

Il moto del polo - 2

Per quanto riguarda lampiezza del periodo

pseudo-euleriano, le osservazioni danno circa

0.3, cioè circa 9 metri sulla superficie.

La figura mostra la posizione del polo di

rotazione dal 1996 al 2000, secondo le

osservazioni VLBI, più una re-analisi della

posizione del polo convenzionale a partire dal

1890. Lasse X è diretto verso Greenwich, lasse

Y a 90 verso Ovest.