Elementi di Astronomia e Astrofisica per il Corso di Ingegneria Aerospaziale Parte II

1
Elementi di Astronomia e Astrofisica per il Corso
di Ingegneria AerospazialeParte II
I movimenti dell'osservatore terrestre Precessione
degli equinozi Nutazione Nutazione libera o
euleriana (il moto del Polo) L'aberrazione e la
deflessione gravitazionale della luce Il tempo
in astronomia Esercizi
Avvertenza questa parte è in larga misura
opzionale, ed è stata svolta molto parzialmente a
lezione. Basta leggerla e ritenere gli elementi
fondamentali effettivamente svolti.
2
I movimenti dellosservatore terrestre
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 in respect to the fixed stars will
result in a time variation of the coordinates.
The stars ideally provide a fixed reference
frame among which well 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.
3
Prime considerazioni dinamiche
The equatorial system (?, ?) is the only 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 in 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. 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. More rigorously, it is the
Earth-Moon (E-M) barycenter that follows Keplers
laws (although 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.
4
Orientazione e rotazione della Terra - 1
The distribution of the masses is far from
spherical or simply azimuthal symmetry, the
instantaneous rotation axis is influenced by the
presence of the Moon and of the Sun (precession
and nutation), and finally the instantaneous
rotation axis doesnt necessarily coincide with
the minor axis of the geometrically best-fitting
ellipsoid (free or Eulerian nutation). Due to
all these factors, the instantaneous rotation
axis passes through the barycenter, but its
variable orientation with respect to the fixed
stars causes 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
20m diameter.
5
Orientazione e rotazione della Terra - 2
Further, the Earth is not perfectly rigid, and
the distribution of masses can change both at the
surface (winds, tides, currents) and in the
interior (earthquakes). The mass of the whole
Earth is M? 5.976x1027 g, the mass of the ocean
is some 10-4 M? , and that of the atmosphere
about 10-7 M? . 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. To be
sure, all these complications are small, even
negligible for many astronomical applications,
but they are well measurable with todays
instruments.
6
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, as
explanation for the increase of longitudes, a
rotation in direct (anti-clockwise) sense of the
whole sphere of the fixed stars around the
ecliptic pole. The Sun would therefore encounter
point ? 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 made 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).
7
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 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- ?).
Two equivalent 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 to the small circle
distant ? from the ecliptic pole and to the
equinoxial colure.
8
La posizione dei poli celesti
During this movement, the celestial poles will be
seen in different constellations today, the
North pole approaches the bright star ? UMi. 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 being is ? Octanctis, of
visual magnitude 5.5).
9
La spiegazione di Newton (1687)
The phenomenon described by Copernicus was
explained on a dynamical basis by Newton in his
Principia (1687) the figure of the Earth must be
taken explicitly into account, and it cannot be a
perfect sphere (more precisely, the internal mass
distribution cannot have spherical symmetry). The
figure can be described as a spheroid having
equatorial axis larger than the polar one. In
this way the Moon, that usually is 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 of the
Moon (quite 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). The Sun exerts 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 more than twice that of the Sun (as
well prove in more detail). Therefore the term
luni-solar precession is usually employed.
10
La nutazione
Copernicuss description, and Newtons
explanation must be only partially true, for two
reasons first, the Moons orbit is inclined by
59 to the ecliptic plane, second, the distance
of the Moon and (to lesser extent) that of 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 has an oscillatory movement
(that Bradley called nutation, the same term used
for the oscillation of a ships masts) of
amplitude ? 9 and period 18.6 years around a
mean position. The period is exactly the same of
the retrogradation of the nodes of the lunar
orbit on the ecliptic.
11
Il moto delleclittica
We have till now 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. At the same
time, 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 present to
have a secular decrease of about 0.5 per year
from the times of Copernicus to the present day,
? has therefore considerably decreased. As 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.
12
I movimenti dei piani fondamentali - 1

• Consider in the Figure the two fundamental planes
  at two dates t1 and t2 (with t2 later than t1 in
  order to fix the sense) intersecting in ?1 and
  ?2, with obliquities ?1 and ?2 respectively. Each
  element, e.g. angle J, can be thought of as
  composed of two parts
• a secular one
• a short period one.

13
I movimenti dei piani fondamentali - 2
During several decennia, or even centuries, the
first part can be developed in a time series with
only the first few terms of importance, so we can
legitimately write
where the secular terms (precession) are zero at
the initial epoch, but not the short period ones
(nutation). The instantaneous elements are the
true elements (true equator, true equinox, true
obliquity, etc.) Those freed by nutation, namely
affected only by the lunisolar precession, are
the mean elements (mean equator, mean equinox,
mean obliquity, etc.)
14
Effetti al primo ordine della precessione - 1
Well 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
(per tropical year)
where ? is the notation favored by many authors,
and that well also use in the following 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
its value derive from the Moon and 1/3 from the
Sun.
15
Effetti al primo ordine della precessione - 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 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
16
Precessione 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)
17
Precessione 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
18
Precessione 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. Lets 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.
19
La precessione planetaria
Lets 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.
20
La diminuzione dellobliquità ? - 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
21
La diminuzione dellobliquità ? - 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.
22
Precessione in RA e DEC dopo il 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. Attenzione lunità di
tempo non è più lanno tropico, ma lanno
giuliano! (tutttavia il rapporto tra anno
giuliano e anno tropico è appena 1.00002136, e da
solo non giustifica il valore della forte
revisione di G).
23
La nutazione - 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).
Lets 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.
24
La nutazione - 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.
25
La nutazione - 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
26
La nutazione - 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.
27
La nutazione - 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
28
La nutazione - 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) . 
29
Precessione più nutazione
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.
30
Gli angoli di Newcomb- 1
Lets 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
31
Gli angoli di Newcomb - 2
The elements (?A, ?A, zA) are given by
(T in Julian centuries since J2000.0)
The same transformations and 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
32
Precessione e angoli di posizione
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.
33
Alcune considerazioni dinamiche -1
Si può dimostrare che il potenziale
gravitazionale dovuto a Sole e Luna e
responsabile per i movimenti di precessione e
nutazione si può esprimere con
sempre che la Terra sia uno sferoide di
rivoluzione omogeneo, con momenti di inerzia
equatoriale A e polare C, e dove intervengono le
masse, distanze e declinazioni dei due corpi
esterni Per tale sferoide si ha
34
Alcune considerazioni dinamiche -2
In virtù della terza legge di Keplero, se diciamo
P il periodo e n il moto medio
(essendo Q il generico corpo esterno) otteniamo
Dato che la massa del Sole eccede grandemente le
altre due, rimane la sola massa della Luna, che
quindi si può stimare dalle costanti di
precessione e nutazione (il rapporto tra la massa
lunare e quella terrestre vale circa 1/80).
35
Confronto tra il potenziale precessionale e
quello di rotazione
Lets 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! Well examine the
rotation of a free Earth, as if the Moon and
the Sun were not forcing the precession and
nutation.
36
La rotazione della Terra libera - 1
La trattazione di questo problema, dovuta
essenzialmente a Eulero, è piena di difficoltà,
per cui citiamo solo alcuni risultati.
La figura mostra un riferimento XYZ fisso con la
figura della Terra, e quindi rotante nel
riferimento inerziale con velocità angolare ? che
possiamo scomporre nelle 3 componenti ?(?1, ?2,
?3). Lenergia di rotazione vale
dove ? è la velocità diurna (siderale). La
posizione del riferimento rotante in un
riferimento inerziale (il piano XY si può
identificare con lequatore terrestre, il piano
inerziale XoYo con leclittica fissa, i due piani
si intersecano lungo la linea dei nodi passante
per i punti vernali) si specifica con 3 angoli
(?,?,?), detti angoli di Eulero, le cui derivate
si possono esprimere in funzione di (?1, ?2, ?3).
La direzione Z sarà quella del polo della figura
terrestre.
37
La rotazione della Terra libera - 2
Dopo vari passaggi, si può arrivare alle
relazioni
con ?, q incognite costanti iniziali. Daltra
parte, il momento angolare totale M
è costante nello spazio inerziale, anche se la
sua direzione non coincide necessariamente con
quella di ? tuttavia, si può dimostrare che M, ?
e Z (la direzione del polo di figura), rimangono
sempre nello stesso piano. Quindi losservatore
rotante (rigidamente collegato con la Terra) vede
i vettori ? e M ruotare attorno allasse Z con
velocità angolare ? la componente di ? nel piano
equatoriale avendo ampiezza costante ?.
38
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.
39
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.
40
Laberrazione della luce
As seen in the previous chapters, precession and
nutation are phenomena due to the variable
orientation of the observers reference system in
respect to the system of fixed stars. Aberration
instead is an effect due to the finite velocity
of light, and to the varying motion of the
observer in respect to the celestial source.
Whilst the finiteness of the velocity of light
(indicated as usual by c) was suspected by many
philosophers and physicists (Galileo Galilei had
suggested a method to measure it, but he probably
never carried it out), it was Oleg Roemer, an
assistant of J. D. Cassini in Paris, who got a
first reliable indication of its high value, by
using purely astronomical means. Finally, in 1727
G. Bradley discovered on ? Dra (a star not too
distant from the ecliptic pole) the effect of
this finite velocity as a periodic variation of
the apparent coordinates measured at successive
dates by the terrestrial observer, during the
yearly revolution around the Sun. The velocity
that could be derived from these astronomical
observations was confirmed around 1850 by Fizeau
and Foucault.
41
Laberrazione solare - 1      
The Earth describes around the Sun an orbit that
for the moment is taken as circular, with radius
a 1 AU and with uniform velocity vector V,
whose direction is perpendicular to the radius
vector and whose modulus is given by
being P the sidereal year, and n (? 3548/day)
the so called mean motion. The light will cross
the AU in time ?a (a quantity referred to as
aberration time, or equation of light, or light
time for the unit distance). For the geocentric
observer, in those 8 minutes the Sun will have
moved from its apparent position (when the light
left it) to the geometrically correct but
unobservable position corresponding to the
arrival of the light. The angular distance among
the two positions is
(radians), or K ? 20.6)
42
Laberrazione solare - 2      
Teoria completa tenendo conto della ellitticità
dell'orbita
e ? 0.0167 a è il semi asse maggiore
dell'orbita terrestre, P è il perielio
43
Laberrazione solare - 3
La costante di aberrazione solare è allora più
propriamente
The component Vt is responsible for the so-called
elliptical aberration Ke ? 0.343 ? 0s.023, which
changes from day to day and in principle is
observable through the equation of time (see
Chapter 10). In total, the difference in ecliptic
longitude between the aberrated (observable) and
the geometric Sun is
being ?? the longitude of the perigee (at 180
from the longitude of the perihelion at the
present epoch, ?? ? 18h48m). A corresponding
equation must be applied to the Right
Ascensions.
44
Laberrazione annua - 1
The annual aberration affecting the stars was
discovered by Bradley by observing with his
meridian circle the second magnitude star ? Dra,
in an attempt to measure its parallax. During the
year, the Declination of the star oscillated by
some ?20.5 around a mean position, reaching the
maximum deviation at the solar opposition or
conjunction. The motion was too large to be
attributable to a distance effect, and the dates
were three months out of phase with those
expected from the annual parallax furthermore,
Bradley was struck by the close numerical
coincidence with the solar aberration constant,
and thus suspected that the cause was the same,
namely the finite velocity of light. Obviously
the apparent Right Ascension of the star had to
be affected in the same way, but that could not
be measured by Bradley due to the scarce
precision of his clocks.
45
Laberrazione annua - 2
a series of positions of ? Dra from 1920 to 1941,
during a complete revolution of the nodes of the
lunar orbit. One can see a small precessional
effect (small because of the proximity of the
star to the ecliptic pole), the nutation
discovered by Bradley himself (the sinusoid with
period 18.6 years), and finally the annual
variation due to aberration. Bradleys discovery
conclusively proved the correctness of Roemers
hypothesis, gave a direct way to determine K and
a second, and much more precise, way to determine
c.
46
Laberrazione annua - 3
An intuitive way of understanding and measuring
the yearly aberration, based on the Galilean
transformation of velocities, is the following
(see Figure ) let C be the center of the
objective of the telescope, E the intersection of
the optical axis with the focal plane, so that
line EC is the direction of sight. The Earths
velocity vector V points toward an instantaneous
direction named apex of motion. Be ? the
geometrical angle between the line of sight and
the apex, in the plane defined by the two
directions. During time ?t employed by the light
to travel distance CE, the Earth moves by V?t
V/c?CE .Therefore the telescope must be pointed
in direction ?, not ?, inclining it toward the
direction of the apex. From the figure, it is
easily seen that
being the component of the Earth velocity
perpendicular to the apparent direction of the
star.
47
Laberrazione annua - 4
Notice that the effect is independent from the
wavelength, from the focal length of the
telescope, from the distance of the star, and
also from its velocity in respect to the
terrestrial observer the aberration will be
absolutely identical for both components of a
binary system of stars or of galaxies (apart that
due to the slight difference in relative
positions). One could also wonder what is the
correct value of c to be used in observations
with ground telescopes, if that in air or that in
vacuum (the two differ in the visible range by
some 67 km/s in normal conditions of temperature
and pressure, a difference well measurable). The
correct answer is the velocity in vacuum, because
the atmosphere partakes of the same translational
motion of the Earth barycenter, and no further
aberration is introduced by its presence (the
atmospheric refraction is one of the main factors
limiting the precision of positional
measurements, including the determination of the
aberration, but this is an entirely different
effect). The observational proof was obtained in
1872 by the Astronomer Royal G. B. Airy, by
filling his telescope with water the aberration
did not change amount.
48
Laberrazione relativistica
La precedente trattazione non è del tutto
corretta, come si vede dalla precedente figura,
in cui il vettore risultante ha modulo c, e
dunque la costruzione viola la Rekatività
Ristretta. Tuttavia la differenza con la teoria
precisa è piccola, essendo dellordine di (V/c)2.
Più precisamente

• as function of the observable ?. Notice the
  dependence of the correction from the sin2?, not
  of ?. Therefore, the elementary formula is
  approximated to terms of the order of (V/c)2 for
  two distinct reasons
• neglect of higher order terms in trigonometric
  expansions
• incorrect transformation rules
•  
• Numerically, the Galilean and relativistic
  expressions give the same results for the annual
  aberration within 0.002.

49
Effetto dellaberrazione annua sulle coordinate
stellari - 1
In the simplifying hypothesis of circular orbit,
during the year V rotates by 360 in the plane of
the ecliptic with constant modulus, and always
pointing to 90 from the Sun consequently, on a
star having ecliptic latitude ?, the yearly
aberration will appear as an apparent elliptical
motion with semi-major axis parallel to the
ecliptic and equal to K, and semi-minor axis
perpendicular to the ecliptic and equal to Ksin?
this ellipse degenerates in a circle at the
ecliptic poles, in a segment on the ecliptic
itself. The star will never be seen in its
geometrical position, except an ecliptical star
twice a year. The dimensions of this ellipse are
the same for all celestial bodies (planets,
stars, galaxies, quasars, etc.), having the same
ecliptic latitude and do not reflect the
ellipticity of the Earths orbit. In other words,
over large angles the aberration will cause a
(small) distortion of the celestial sphere,
distinctively different from precession and
nutation, who are rigid rotations of the sphere.
Any other periodic motion of the terrestrial
observer, for instance the diurnal rotation, or
the motion around the Earth-Moon barycenter, will
cause a corresponding periodic phenomenon of
aberration, suitably scaled for its velocity
value.
50
Effetto dellaberrazione annua sulle coordinate
stellari - 2
The apparent position of the star on this great
circle is S, displaced from S toward T by
amount K T in its turn is on the ecliptic at
90 behind the Sun, whose true (geometric)
longitude at the date is . The smallness of K
allows using plane trigonometry in the small
triangle SSU after simple passages, one gets
During the course of the year, the star traces
the locus
which is the annual aberrational ellipse, whose
semi-major axis is K.
51
Effetto dellellitticità dellorbita
Lets now add the effect of the slight
ellipticity e of the terrestrial orbit, namely
the small and constant velocity component of
amplitude Ke perpendicular to the semi-major
axis. First of all, as already discussed for the
solar aberration, the value of constant K in the
annual aberration must be understood as
Secondly, the effect of the perpendicular
component is the following the geometric
position of the star is not exactly at the center
of the ellipse of aberration, but displaced in
respect to it by 0.343, in a direction whose
longitude is ? ??-90, namely at 90 from the
geocentric longitude of the perigee of the Sun.
This (almost) constant displacement, called
elliptical aberration, when projected in
longitude and latitude, amounts to (E-terms)
52
Aberrazione annua in coordinate equatoriali-1
Ignorando i termini E, e chiamando
le componenti del vettore velocità della Terra, i
cui valori approssimati sono
si ha
53
Aberrazione annua in coordinate equatoriali-2
In order to remove the elliptical aberration, we
must add the (almost) constant terms
for the FK5 and all catalogues based on it. The
corrections therefore take the form
in which C, D depend on the suns longitude and
therefore on date, while c,c,d,d depend on the
coordinates of the star and on the obliquity of
the ecliptic ?. Notice the formal similarity with
the expression of the nutation, although the
physical bases are so different. We have remarked
that the aberration introduces a slight
distortion of the celestial sphere the distance
s between two stars, and their position angle p,
will be altered. To give an order of magnitude,
over an arc of 1 the maximum effect of the
annual aberration is 0s.02/cos ? in ?, 0.3 in ?.

54
Laberrazione diurna
The diurnal rotation velocity will be responsible
for a similar effect, of much smaller amplitude,
and dependent on the geocentric latitude ? of
the observer. Indeed, the diurnal velocity is
approximately 0.46 km/s at the equator (the
angular velocity is
its apex is on the equatorial plane, at 90 from
the meridian and toward East, therefore with
equatorial coordinates
The diurnal aberration ellipse is thus parallel
to the equatorial system, and its smallness
permits to treat the difference apparent
geometric with the first order formulae for an
observer in geocentric latitude ? at distance ?
km from the Earths center, the difference is
55
Aberrazione stellare e planetaria
We have seen that the correction of annual
aberration provides the geometric direction to
the star at the time when the light reaches the
observer no allowance is made for the motion of
the star in the long time interval between
emission and reception. In the case of the bodies
of the Solar System, whose orbits are known with
high accuracy, we can take explicitly into
account the finite time ? of propagation of the
light from the body to the observer. The term
planetary aberration usually means the sum of the
annual aberration (affecting also the stars in
the field surrounding the body) plus the finite
light time (although some authors actually mean
simply the second term, the proper meaning of
planetary aberration is that here described).
56
La deflessione gravitazionale della luce - 1
There is another effect due to the propagation of
light that was non included in the pre-1984
formulae, namely the gravitational deflection of
the light by the Sun. Such deflection was already
foreseen by the Newtonian theory, but with a
value twice as small as that calculated on the
basis of General Relativity (Einstein, 1915).
Dyson, Eddington and Davidson, taking advantage
of the solar eclipse of May 1919, confirmed
(although certainly not in a conclusive way) the
correctness of Einsteins prediction. The
gravitational deflection of light, together with
the precession of the perihelion of Mercury and
the gravitational red-shift of the spectral
lines, is since then one of the fundamental
astronomical proofs of that theory. Karl
Schwarzschild in 1915 introduced a typical radius
associated with a spherical mass M, the so-called
Schwarzschilds radius ?S given by
whose value is about 1.5 km for the Sun and 0.44
cm for the Earth.
57
La deflessione gravitazionale della luce - 2
The influence of the mass of the Sun on a grazing
light ray will make its path slightly concave
toward the Sun this effect is the manifestation
of the curvature of space due to mass. Therefore,
in first approximation, a star near the limb of
the Sun, having radius R, will be seen by the
terrestrial observer in a direction slightly
displaced, radially outward, by the quantity
AU and angular radius of the Sun
The constant Q is equal to 2 in the Newtonian
theory, and to 4 in General Relativity therefore
is 0875 in the first case, 1.75 in the second.
All measurements (see for instance Jones, 1976)
have confirmed, within the errors, the
relativistic value. Notice that the deflection is
independent from the wavelength, it is the same
in the optical and radio domain, and actually the
radio measurements are much more convincing than
the optical ones, having confirmed the validity
of General Relativity to better than 1
58
La deflessione gravitazionale della luce - 3
The outward radial displacement of the apparent
direction of the star decreases linearly with the
angular distance from the center of Sun, so that,
after some manipulation, the general formula can
be easily derived
At really grazing incidence and when the solar
diameter is 0.25, the value of the displacement
is 1.866. Notice that at 45 from the center of
the Sun the displacement is still at the level of
0.01, and of 0.004 at 90. An appropriate
projection of this angle on the equatorial system
will permit the determination of the corrections
to be applied to the apparent coordinates.
59
Il tempo in astronomia
In many considerations of the previous chapters,
time was found necessary to properly describe the
movements of the celestial sphere in respect to
the meridian. Time also enters in the Newtonian
dynamical explanation of the motions, as the
fundamental independent variable in differential
equations. In the present chapter, several
operative definitions of time will be given,
together with the transformations among them.
Well consider four different time scales
sidereal, solar, dynamical, atomic, the first
three being strictly associated with astronomical
observations. Furthermore, in questions where
General Relativity matters, it will be necessary
to distinguish between proper time and coordinate
time, and time will become a component of the
overall space-time geometry.
60
La rotazione diurna
The diurnal rotation takes place around a polar
axis whose direction in respect the distant stars
(namely in an inertial frame of reference) we
consider here as invariable, and with absolutely
constant angular velocity. In other words, this
rotation is expressed as a vector ?, which is not
only constant but also invariably coincident with
the polar axis c of the ellipsoid, which
mathematically describes the Earths
figure. Quale sarà la durata della rotazione
diurna? La possiamo misurare dallintervallo di
tempo tra due passaggi consecutivi di una stella
(ma non del Sole!), equatoriale e priva di moto
proprio, in meridiano. Questo giorno stellare
tuttavia non viene usato. Si usa piuttosto il
passaggio del punto vernale ??, che si sposta
rispetto alla stella di circa 0.008 secondi
/giorno causa la precessione luni-solare. La
differenza tra giorno stellare e giorno siderale
è dunque minima, per cui si può usare
tranquillamente la durata del giorno siderale
(24h di TS) come periodo di rotazione diurna
della Terra. The ratio between the mean sidereal
day and the period of rotation of the Earth at
the present epoch is 0.99999990 it varies very
slowly because of the varying precessional
constant, by about 1 part over 6x1013 each
century.
61
Il tempo siderale
We have already defined the Sidereal Time ST as
the Hour Angle of the equinox ?, ST HA(?). At
each rotation of the Earth, HA increases by one
sidereal day of 24 hours. Notice that HA is an
angle defined on the celestial equator, but the
equinox ? itself is not directly visible as a
point, being actually defined by the declination
of the Sun ?? through the relation
In other words, ST is defined by the Sun, not by
the stars however, this very delicate operation
of referring the equinox directly to the Sun is
seldom done. In order to determine ST, it is much
easier to utilize the upper meridian transit of a
set of fundamental stars (e.g. of the FK5), whose
right ascensions define also the origin of the
system. A word of caution here, because each
particular set of fundamental stars defines a
slightly different vernal equinox. Presently, the
best realization of the fundamental catalogue is
the already quoted ICRS (adopted by resolution of
the IAU starting Jan. 1st, 1998).
62
La non-uniformità del Tempo Siderale
However, this sidereal time is only approximately
uniform even disregarding the irregularities of
the rotation, the position of ? is affected by
nutation, which is composed by a superposition of
many different periodic terms, in particular in
dependence of the longitude of the node of the
lunar orbit. Therefore we have to distinguish
between apparent and mean time the difference
in the sense apparent ST minus mean ST, is
said equation of the equinox (before 1960, EE was
also called nutation in Right Ascension). The
amount of EE is always between ?1s.179, with a
periodicity of 18.6 years. For instance, in 1985
EE was -0s.83 at the beginning and -0s.56 at the
end of the year. The daily variation of the
duration of the sidereal day is therefore about
10-4s, but it accumulates for several years
before changing sign. EE became measurable around
1930, when the precision of the clocks became
better than one millisecond per day since then
we have to distinguish mean from apparent
sidereal time the first is more uniform than the
second, but it is the second that enters in the
telescopic observations.
63
La rivoluzione annua
The annual movement of the Sun in respect to the
fixed stars, of approximately 1 deg per day,
Eastward on the ecliptic, reflects the revolution
of the Earth according to the first two of
Keplers laws  I the orbit is an ellipse
having the Sun in one of the two foci, with
semi-minor axes a and b, having equation
The initial direction is usually taken to
coincide with that of the semi-major axis a, when
the Earth passes at the perihelion P (or the Sun
at the perigee ?), so that the argument (? - ?0)
is replaced by the so-called true anomaly ?.
II the areal velocity (not the angular one!)
is constant
where r is the distance Earth-Sun.
64
Il Tempo Solare - 1
Lets therefore call solar day the interval of
time between two successive upper culminations of
the Sun on the meridian of a particular site, and
solar time T? the Hour Angle of the Sun,
augmented by 12 hours (in this way the solar day
starts at midnight, not at noon this convention
was adopted in 1925, but for the 3 following
years not all Observatories conformed to the
resolution, so that care must be taken when using
the dates preceding 1928)   T? HA?12h
This is the time indicated by a sundial (apart
from the effects of the atmospheric refraction
that can be ignored in this context), in that
particular place. However, the Sun as a
geometrical point does not belong to the equator,
but to the ecliptic, moving on it according to
Keplers first two laws those two factors affect
both the duration and the uniformity of the solar
time.
65
Il Tempo Solare - 2
Indeed, the Sun appears to move in direct sense
(Eastward) on the ecliptic by approximately 1
each day (more precisely, by 360/365 days
3m56s/day) in respect to the fixed stars, and
therefore also in respect to the equinox (at
least in this approximation) this is the extra
time the Sun takes to pass the following day in
meridian in respect to the equinox. The solar day
is then, on the average, 3m56s longer than the
sidereal day, and equally all units of solar time
are correspondingly longer than the units of
sidereal time having the same name. The above
considerations are only very roughly true, the so
defined solar time is grossly non-uniform, as
well show in a moment. Notice that while the
sidereal time finally derives for the rotation of
the Earth, the solar time has two independent
causes, namely the diurnal rotation and the
yearly revolution those two movements do not
have any fundamental connection (apart a very
slight influence through the constants of
precession that we may safely ignore here) this
independence is also at the root of the
difficulties in building calendars based on the
solar day and on the solar year.
66
La non-uniformità del Tempo Solare - 1
Given that T? HA?12h , HA? ST -
??12h  we understand that the non-uniformity of
T? is the same of that of ??. Let ??, ??, ?? be
respectively the ecliptic longitude, right
ascension and declination of the Sun the
following relations can be easily derived
Taking the time derivative of the second and
inserting the first we also get
which comprises both the above mentioned effects,
that of projection on the equator and that of
variable angular velocity on the ecliptic.
67
La non-uniformità del Tempo Solare - 2
To quantify the non-uniformity of we must take
into account that         Keplers II law
insures that the areal, not angular, velocity is
constant therefore the Sun has a daily motion
greater at the perigee than at the apogee
around the second of January
around the 2nd of July
        The same motion of
on the ecliptic, is projected on the equator on
different arcs according to the declination, from
a minimum value of cos? (? 3m37s) per day at the
equinoxes to a maximum value of 1/cos? (? 4m16s)
at the solstices. 
68
La non-uniformità del Tempo Solare - 3

• Therefore the duration of the true solar day is
  continuously variable more precisely, the
  longest solar day happens around mid-December,
  and lasts about 24h00m30s, some 53s longer than
  the shortest day around the autumn equinox. Those
  seemingly small differences steadily accumulate
  with the passage of the days, reaching several
  minutes before changing sign, as well discuss
  later (see Equation of Time).
• In order to construct a truly uniform solar time,
  lets introduce, following Newcomb, two
  hypothetical Suns with uniform motion
• - a fictitious one F? on the ecliptic (called by
  some authors Dynamic Mean Sun), which coincides
  with the true Sun at perigee and apogee
• - a mean one M? on the equator that encounters F?
  at the equinoxes
• Both bodies move with the same daily motion,
  per day, which derives from the
  length of the tropical year (period of time
  between two consecutive passages of the Sun
  through point ?).

69
Lequazione del centro - 1
The difference
it referred to as equation of the center EC. It
can be calculated from the equation of motion of
the Sun in its orbit. Leaving the demonstration,
it will be sufficient to affirm that, e being the
eccentricity of the orbit, t0 the instant of
passage of the Sun for ? (around the 2nd of
January, when ?(?) ? 282), ?(t) the true
anomaly, M(t) n(t - t0) an auxiliary quantity
uniformly increasing with time and said mean
anomaly, the following relations obtain
70
Lequazione del centro - 2
The equation of the center EC is therefore a
periodic function of time, with period of 12
months and amplitude of about 115, namely 7m40s,
roughly corresponding to 2 solar diameters. The
phenomenon is so evident that already Claudius
Ptolomeus could ascertain it, although with an
excessive value. We credit Copernicus with a
determination very close to the true one. Lets
now take the derivative of EC
Given that M 0 the 2nd of January, 90 the
3rd of April, 180 the 2nd of July, 270 the
1st of October, we can easily calculate the
variation of angular velocity at each date.
Notice that
only in two occasions, when the Sun passes
through the semi-minor axes of its orbit.
71
Lequazione del tempo - 1
Dalla relazione
che è lequazione trascendente già discussa nel
Cap. I, deriviamo
When the Fictitious Sun F? encounters the equator
at ? coming from ? (some time after the true
Sun), let the Mean Sun M? start from ? with the
same uniform motion n. The two hypothetical Suns
will coincide again in ? in this way, at any
instant
72
Lequazione del tempo - 2
Finally, calculate the equation of time E, namely
the difference
E its a fairly complex function of time (see
Fig. 4. 1). Its value is zero at four times a
year, namely at the beginning of April, middle of
June, beginning of September, around Christmas
the maximum value of about 16m is reached in
early November, the minimum value of 14m in
middle February. Notice that the exact values at
a particular date will vary by few seconds from
one year to the next, in a periodic behavior due
to the presence of the leap (in Latin, bisextus)
year.
73
Il Tempo Universale UT
For any particular site, the difference between
the right ascension of the true and Mean Sun will
also equal the difference, changed in sign,
between their two Hour Angles HA?- HA(M?)
-?? ?(M?) E The Hour Angle of the Mean Sun,
augmented by 12h in order to have the day start
at midnight, is called the local mean solar time
T(M?)
In particular, the Mean Solar Time at Greenwich
is called Universal Time UT. The interval of time
between two passages through the local meridian
of the Mean Sun is properly called Solar Mean Day
(indicated with j), and it is divided in 24h,
each of 3600 seconds of mean time (whose length
is not the same of the sidereal second).
74
I differenti ritmi
By definition therefore, the sidereal time ST and
the Mean Solar Time T(M?) have the same degree of
uniformity of the Earths rotation, but they
differ both in rate and origin. The constant
ratio between the two rates can be easily
determined. Lets call tropical year the interval
of time between two consecutive passages of the
Mean Sun through the vernal equinox, a quantity
determined in mean solar days with utmost
precision thanks to its recording over several
millennia apart a slight secular variation due
to changes in the constants of precession,
Newcomb found   1 tropical year 365.2421988 j
365j05h48m45s.975 366.2421988 sidereal
days   because after 1 tropical year one more
sidereal day will have elapsed. Therefore  
rate ST (1 1/365.2421988 1.002737909)
rate T(M?) rate T(M?) (1 -
1/366.2421988 1-0.002730434 0.997269566) rate
ST 24h T(M?) 24h3m56s.55537 ST , 24h STS
23h56m04s.09053 T(M?) 1s T(M?) 1s.0027379
ST , 1s ST 0s.9972696 T(M?)
75
Relazione tra UT e ST - 1
For another site at longitude ?, expressed in (h
m s), T(M?) UT ? ?, where the sign is if East
of Greenwich, - if West. ? being an angle, it is
absolutely equivalent to express the difference
in longitude between two sites as difference in
solar or sidereal time. Lets discuss the
origins of the two times. According to Newcomb,
the mean longitude of the non-aberrated F? at 12h
UT (noon) of Jan. 1st, 1900 had the
value   ?(F?) 280?4056.37
18h42m42s.391   At the same instant, that was
also the ST at Greenwich. Notice that the
non-aberrated Sun, not the apparent one, which is
20.45 behind it, enters in this definition.
76
Relazione tra UT e ST - 2
After a whole Julian year of 365j.25, the value
of ST augments by 86 401s.845 (1 day in a
tropical year, plus the difference corresponding
to 0.0078 days), plus the minute acceleration of
the precessional constants. Using the current
values of the constants, and counting the time T
in Julian centuries since Jan. 1st, 2000 at 12h
UT (therefore from noon, not from midnight!), the
complete expression of the mean ST at the
midnight of Greenwich, at any date T
is STGreenwich (0h UT) 6h41m50s.5481 8 640
184s.812866?T 0s.093104?T 2 - 6s.2?10-6?T 3
where the last two terms derive from the
variation of the precessional constants, and the
time T should be expressed in the scale UT1 well
discuss later.
77
Lanno - 1

• The yearly revolution of the Earth permits the
  definition of a new time scale, and of a new unit
  of time, namely the year, in several different
  ways
• Anno tropico intervallo di tempo tra due
  passaggi del Sole per ?
• 1 tropical year 365j.24219879 0j.00000614?T
  se lorigine dellanno tropico è fissato
  allistante in cui la longitudine del Sole
  fittizio è 280, si ha lanno besseliano B
• - The sidereal year is the interval of time
  between two passages of the sun over an ecliptic
  star devoid of proper motion. Therefore the
  sidereal year is longer than the tropical one by
  the amount of the precession of ? along the
  ecliptic, namely by approximately
  (1296000-50.3)/1296000, corresponding to
  20m24s, or else to 35000 km along the orbit of
  the Earth. Therefore the duration of the sidereal
  year is of 365j.25636. This value is not
  measured, but derived from the length of the
  tropical year. From it, we get also the mean
  solar motion
• n 1296000''/365j.25636 3548''.1928''/j
• whose value is not affected by the secular
  variation of the precessional constant, and
  therefore has the same uniformity of the diurnal
  rotation.

78
Lanno - 2

• The anomalistic year is the interval of time
  between two different passages of the Sun through
  the perigee. The direction of major axis of
  Earths orbit (or in other terms, the line of the
  apsides) however is not fixed in the inertial
  space, it slowly precesses in the same direction
  of the yearly motion, by an amount of 11.63/year
  that is essentially determined by the
  gravitational perturbations of the other planets,
  plus a much smaller contribution due to General
  Relativity (sometimes referred to as geodesic
  precession). Therefore the longitude of the
  perigee, referred to the moving equinox,
  increases of about 11''.6 50''.26
  61''.89/year.
• The anomalistic year is longer of the previous
  ones, its duration being of approximately
  365j.25964, with a secular acceleration of
  0.263s/century. It is easily seen that perigee
  and equinox coincide every 21000 years because
  the duration of the seasons depends from the
  distance between equinox and perigee, the
  consequence is their appreciable variation, at a
  level of one hour per century.

79
Lanno - 3
We quote two more years, the draconic (or
draconitic) and the Gaussian. -