Lezioni Terza Settimana

1
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!
2
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.
3
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.
4
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.
5
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.
6
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.
7
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.
8
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).
9
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.
10
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).
11
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.
12
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.
13
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.
14
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.
15
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.

16
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.)
17
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.
18
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
19
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)
20
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
21
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.
22
A graphic representation of the first order
precession
In RA
In DEC
23
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.
24
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
25
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.
26
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).
27
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.
28
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.
29
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
30
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.
31
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
32
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) . 
33
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.
34
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
35
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
36
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.
37
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.
38
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.
39
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.
40
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.
41
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).
42
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 ?.
43
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.
44
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.