31 01 05C. Barbieri Elementi_AA_ 2004- 05_Terza settimana 1 Lezioni Terza Settimana Esercizi in aula...

31 01 05 C. Barbieri Elementi_AA_ 2004-05_Terza settimana

1

Lezioni Terza Settimana

Esercizi in aula (non riportati qui)I movimenti dell'osservatore terrestrePrecessione degli equinoziNutazioneNutazione libera o euleriana (il moto del Polo)

Questa parte è quasi tutta scritta in inglese!


2

Tempo e anno sideralePremettiamo 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 - 2In 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 Kepler’s laws (it may be worth recalling that even in the absence of the other planets, the E–M 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 doesn’t 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 - 2The 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 today’s 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 Earth’s 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 - 2The 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 Earth’s 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).

V W 3S

' ' ' cosM

f f f K ar

€€

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 Earth’s 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 nutationCopernicus’s description, and Newton’s explanation, must be only

partially correct, for two reasons: first, the Moon’s orbit is inclined by 5°9’ 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 ship’s 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 eclipticUntil 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

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: 1. a secular one (actually a periodic one but with very long period)2. a periodic one, with longest period equal to 18.6 years, and many

shorter ones.

Notice: the movement of the equator is much larger than that of the ecliptic!


16

The movements of the fundamental planes - 2

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.)

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:

2 30( ) ( ) ( )J t at bt ct n t n t


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:

50".37 per tropical year

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 - 2In 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 P+dP 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:

dd d 50".37

sin

Jt


19

Precession in AR e DEC - 1The projection of this elementary arc on the moving equator amounts to the elementary luni-solar precession in Right Ascension:

dd d cos cos 46".21

sin

Jt

while the perpendicular component along the Hour Angle of 1 is the elementary

lunisolar-precession in Declination:

=d ( d )cos sin 20".05 n J t

Returning to angular velocities, it is customary to put:

cosm (m = +46”.21/y= +3s.08/y)

sinn (n = +20”.34/y = +1s.34/y)

(where y is the tropical year)


20

Precession in AR e DEC - 2Taking the time derivative of the transformation between ecliptic and equatorial coordinates:

d , d

and with the initial assumptions that:

from the transformation between equatorial and ecliptic coordinates we easily derive:

0, 0

-1

(cos +sin sin tan )

sin tan [3 .08 1 .34sin tan ] ys sm n

-1sin cos cos 20".05 cos yn


21

Precession in RA e DEC - 3

Notice however that d/dt becomes very large in the proximity of the celestial poles, so that the calculation there becomes critical.

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 B1950.0+t. In a first approximation, the following formulae will suffice:

1950 1950 1950 1950 1950 1950( sin tan )t m n t

1950 1950 1950 1950sint n t


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:

-10".13 yg

that must by subtracted from the value of the luni-solar precession to obtain the value of the general precession: -1cos +50".26 yG g

As a consequence, the value of constant cosm g slightly decreasesto about 46”.07 y-1 = 3s.07 y-1. The precession in Declination is not affected.


24

The diminution of the obliquity - 1There 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:

A Asin sin sin sin( )g

Acos cos sin cot sin cot( )g g


25

The diminution of the obliquity - 2A, , g are very small angles, so that:

A Asin cosecg -1A Acos 0".47 y

After years, the inclination of the mobile ecliptic will be:

A A/ cos

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 = 23°30’, while today is measured closer to 23°26’. As a consequence, none of the ‘constants’ , m, n, G, g, A, A, is really a constant.


26

Precession in RA and DEC – after 1984Since 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 , = 23°26'21''.448 - 0''.00468150 Tm = 46''.124362 + 0''.02793 T , n = 20''.043109 - 0''.008533 T

0".46815 0.0007T cos 0".1055 0".0189g 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 - 1We 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 = 5°9’ 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 - 2Correspondingly, 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:

( ) 250 10 '50".37 19 .34145336N t

if t in tropical years since 1900.0, or also:

2( ) 125 .04452 19 .341363 0 .00002971N t t

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 - 3Consider 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:

sinsin ' sin

sin

iN N

i sin

' arcsin( ) 13sin

iN

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 Bradley’s measurements:

17".2sin ( )N 9".2cos ( )N


30

The nutation - 4This 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:

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 ratio of the two axes was justified on dynamical reasons by d’Alembert, who showed that it must be equal to cos2 /cos.


31

The nutation - 5The 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:

(cos sin sin tan ) (sin cos )

The term in obliquity causes a variation of , but it doesn’t affect ; after few simple calculations we obtain:

cos tan sin


32

The nutation - 6and in total:

(cos sin sin tan ) cos tan

sin cos sin

For instance, for the year 2000 and to a precision of 1”, the Astronomical Almanac gives the following numerical expressions:

0 0048sin(125 .1 0.053 ) 0 .0004sin(198 .0 1.971 )

0 .0026cos(125 .1 0.053 ) 0 .0002cos(198 .0 1.971 )

d d

d d

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:

( sin tan ) cos tan 'A m n B E

cos sinAn B

A, B, E’ being function of the date named Bessel’s 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- 1Let 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:

A 0 0 A

A A 0 0 A A 0

A 0 0 A A 0

cos sin( ) cos sin( )

cos cos( ) cos cos cos( ) sin sin

sin sin cos cos( ) cos sin

z

z


35

Newcomb's precessional angles- 2The elements (A, A, zA) are given by:

2A

2A A

2A

0 .6406161 0 .0000839

0 .000202

0 .5567530 0 .0001185

T T

z T

T T

(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:

11 A A A A Asin sin cos cos cosP z z

To allow for nutation, the rotation P will be followed by rotation RN:

N

1 cos sin

cos 1

sin 1

R


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 (they’ll 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:

/ sin secp t n

In turns, the differential coordinates of the two nearby objects (1 2

, 1 2 ) will change by:

dsec cos sin sin sec

dn

t

dsin

dn

t

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 -1It is possible to demonstrate that the gravitational potential due to Sun and Moon, and responsible for the precession and nutation is:

2 2prec 3 3

( )3 sin sin

2

M MC AU G

r r

‚ €

‚ €‚ €

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:

2

2 2

21.003

aC

A a c

2 21( )

5A M b c

2 21( )

5C M a b a b a A B

Notice the presence in Uprec of the declination of the two bodies.


38

Dynamical considerations -2Using 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:

2 22 3

2Q

4 4( ) ( )P M M M M a

n G

‚ ‚ ‚

2 2 2 2prec

( )3 sin sin

2 ( )

MC AU n n

M M

‚‚ ‚ € €

‚

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:

2 22 3

2Q

4 4( ) ( )P M M M M a

n G

€ € €

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:

21

2T C -5 -12 / (sidereal day) 7.292 10 s

Uprec can be maximized by taking the maximum value of the declination of Sun

and Moon, whence: 22

( )3 1 0.25precU n M nC A

T C M n

€ ‚ ‚

€

/ 1/13 , / 1/ 366.25n n n ‚ € €7

prec / 1 10U T

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 - 1The 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:

2 2 2 2 2 21 2 3 1 3

1 1 1( ) (2 )

2 2 2T A B C A C C

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 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 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.


42

The rotation of the free Earth - 2After several passages we obtain the relations:

1

2

3

cos

sin

C At q

A

C At q

A

with , q unknown initial constants. On the other hand, the total angular momentum M:

1 2 3 1 1 2 2 3 3( , , ) ( , , ) ( )M M M I I I dm M M R Ω R

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 - 1Per l’osservatore terrestre dunque, il polo di rotazione diurna sembrerà descrivere un moto di precessione attorno al polo di figura (moto detto di polodia), con frequenza:

pol 0.003C A

A

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) sull’orizzonte, 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.


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Il moto del polo - 2Per quanto riguarda l’ampiezza 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. L’asse X è diretto verso Greenwich, l’asse Y a 90° verso Ovest.

31 01 05C. Barbieri Elementi_AA_ 2004- 05_Terza settimana 1 Lezioni Terza Settimana Esercizi in aula...

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Transcript of 31 01 05C. Barbieri Elementi_AA_ 2004- 05_Terza settimana 1 Lezioni Terza Settimana Esercizi in aula...