Lezioni IV settimana

1
Lezioni IV settimana
L'aberrazione della luce La deflessione gravitazi
onale della luce Il tempo in astronomia Stagioni
, calendario Esercizi Anche questa parte è in
inglese!
2
The aberration of light
As seen in the previous chapters, precession and
nutation are phenomena due to the variable
orientation of the observers reference system
with 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 with 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.
3
The solar aberration - 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 t
he AU in time ?a ? 8 minutes (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
those two positions is
(radians), or K ? 20.6
4
The solar aberration - 2      
We can easily derive a more accurate description
by taking into account the elliptical shape of
the orbit
e ? 0.0167 is the small ellipticity
a is the semi-major axis The point P is th
e perihelion. The Earth's velocity vector V ca
n be decomposed in a component perpendicular to
the radius vector, and one perpendicular to the
semimajor axis
Notice that Vt is constant not only in amplitude
but also in direction!
5
The solar aberration - 3
The constant of the solar aberration is then more
properly
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 a
later paragraph). In total, the difference in ec
liptic longitude between the aberrated
(observable) and the geometric (unobservable) Sun
is
being ?? the longitude of the perigee (at 180
from the longitude of the perihelion at the
present epoch, ?? ? 18h48m). A corresponding equ
ation must be applied to the Right Ascension
difference.
6
The annual aberration - 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 approximately ?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 o
f 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.
7
The annual aberration - 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.
8
The annual aberration - 3
An intuitive way of understanding and measuring
the yearly aberration, based on the Galilean
transformations 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 EC, the Earth moves by V?t
EC?V/c. Therefore the telescope must be pointed
in direction ?, not ?, inclining it toward the
direction of the apex. From the figure, it is ea
sily seen that
being the component of the Earth velocity
perpendicular to the apparent direction of the
star.
9
The annual aberration - 4
Notice that the effect is independent from
i) the wavelength, ii) the focal length of the
telescope, iii) from the distance of the star,
and also iv) from its velocity with 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 co
rrect 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
approximately 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.
10
The relativistic aberration
The previous treatment is not entirely correct
the figure shows that the resultant vector has
modulus c, thus violating the Special Relativity
restrictions. However, the difference with the
correct theory is small, of the order of (V/c)2.
More precisely

• 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.

11
Effect of the annual aberration on the stellar
coordinates - 1
In the simplifying hypothesis of circular orbit,
during the year V rotates by 360 in the plane of
the ecliptic with constant modulus, 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.
12
Effect of the annual aberration on the stellar
coordinates - 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.
13
Effect of the ellipticity of the orbit
Let us 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 with
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)
14
Annual Aberration in equatorial coordinates -1
By ignoring the E-terms, and calling
the equatorial components of the Earth's velocity
vector, whose approximate values are
(in AU/day, c 173.14 AU/day) we have
15
Annual Aberration in equatorial coordinates -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 the 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
angular 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 ?.
16
The diurnal aberration
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, while 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
17
Stellar and planetary aberration
We have seen that the correction for 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 bodi
es 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. By
consequence, we shall form the astrometric
position of a body of the Solar System by
applying the correction for the barycentric
motion of the body during the light time to the
geometric geocentric position referred to the
equator and equinox of the standard epoch of
J2000.0. Such a position is then directly
comparable with the astrometric position of stars
formed by applying the correction for proper
motion and annual parallax to the catalogue
position for the standard epoch of J2000.0.
18
The gravitational deflection of the light - 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 alrea
dy 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 3.0 km for the Sun and 0.88
cm for the Earth.
19
The gravitational deflection of the light - 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
with radio data) 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
20
The gravitational deflection of the light - 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.
21
The Time in Astronomy
In many considerations of the previous chapters,
time was found necessary to properly describe the
movements of the celestial sphere with 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.
We shall consider four different time scales
sidereal, solar and dynamical times,
atomic time, the first three being strictly asso
ciated with astronomical observations, the fourth
with the terrestrial laboratory.
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.
22
The diurnal rotation
The diurnal rotation takes place around a polar
axis whose direction with 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. Which is the duration of the diu
rnal rotation? We can measure it by the interval
of time between two consecutive upper transit in
meridian of an equatorial star, devoid of proper
motion (but not of the Sun!). This stellar day
however is not used. Instead astronomers use the
interval of time between two upper transits of
the vernal point??, which is in motion with
respect to the ideal equatorial star by
approximately 0.008 seconds/day because of the
luni-solar precession. This difference is so
minute that in many application we can use the
duration of the sidereal day (24h of ST) as the
period of the diurnal rotation of the Earth. 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.
23
The sidereal time
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
Notice, in the spherical triangle one side might
be larger than 180
In other words, ST is defined by the Sun, not by
the stars. This very delicate operation of refer
ring 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).
24
The non-uniformity of the Sidereal Time
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 m
ean 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.
25
The annual revolution
The annual movement of the Sun with 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 havin
g 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!) i
s constant
where r is the instantaneous distance Earth-Sun.
26
The Solar Time - 1
Let us 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 geometr
ical 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.
27
The Solar Time - 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) with respect to the fixed stars, and
therefore also with respect to the equinox (at
least in this approximation) this is the extra
time the Sun takes to pass the following day in
meridian with 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 we 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.
28
The non-uniformity of the Solar Time - 1
Given that T? HA?12h, HA? ST - ?? , T?
ST - ?? 12h we understand that the non-uniform
ity of T? is the same of that of ??
(disregarding in the present context the minute
accelerations of ??). Let ??, ??, ?? be respectiv
ely 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.
29
The non-uniformity of the Solar Time - 2
To quantify the non-uniformity of we must take
into account that         Keplers II law insur
es 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. 
30
The non-uniformity of the Solar Time - 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, approximately 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,
let us 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 sam
e daily motion, which has the value
per day
a value which derives from the length of the
tropical year (period of time between two
consecutive passages of the Sun through point ?).
31
The equation of the centre - 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, being e 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 called mean
anomaly, the following relations obtain
32
The equation of the centre - 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.
Let us 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.
33
The Equation of Time - 1
From the relation
which is a form of the already discussed
transcendental equation, we derive
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
34
The Equation of Time - 2
Finally, calculate the equation of time E, namely
the difference
E is a fairly complex function of time (see
Figure). Its value is zero 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.
35
The Universal Time 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 Angl
e 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).
36
The different rhythms
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 rhythm and origin. The constant
ratio between the two rhythms can be easily
determined. Let us 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 3
65j.2421988 365j05h48m45s.975 366.2421988
sidereal days   (because after 1 tropical year o
ne more sidereal day will have elapsed).
Therefore   rate ST (1 1/365.242
1988 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?)
37
Relation between UT and ST - 1
By definition then
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.
Let us discuss the origins of the two times. Ac
cording 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 18
h42m42s.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.
38
Relation between UT and 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 vari
ation of the precessional constants, and the time
T should be expressed in the scale UT1 we will
discuss later.
39
The year - 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
• Tropical year the interval of time between two
  passages of the Sun through ? 1 tropical year
  365j.24219879 0j.00000614?T if the origin
  of the tropical year is fixed at the instant when
  the longitude of the fictitious Sun is 280, we
  have by definition the Besselian year, indicated
  with 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 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.

40
The year - 2

• The anomalistic year is the interval of time
  between two different passages of the Sun through
  the perigee. The direction of the major axis of
  Earths orbit (or in other terms, the line of the
  apses) 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.

41
The year - 3
We quote two more years, the draconic (or
draconitic) and the Gaussian.
- The draconic year is the interval of time betw
een two passages of the Sun through the ascending
node of the lunar orbit. It is therefore
connected with the occurrence of eclipses. Due to
the retrograde motion of the lunar nodes on the
ecliptic, this year is the shortest, its duration
is of 346j.6201. - The Gaussian year derives fr
om Keplers third law, it is the period of
revolution of a mass-less body in circular orbit
around the Sun at the distance of 1 AU. Its value
is 365j.25690 .
42
The dynamic time

• We discuss now the non-uniformities of Sidereal
  and Universal Times caused by the
  non-uniformities of the diurnal rotation. Let us
  set aside the acceleration due to the secular
  variation of the precessional constant, in order
  to concentrate our attention of the rotation
  itself.
• We can distinguish three types of
  irregularities
  
• A secular slowing down of the rotation, amounting
  to a variation of the mean solar day by about 2
  ms/century, partly but not totally explained by
  tidal dissipation of the rotational energy.
  Because this increase accumulates over the ages,
  the effect on phenomena that took place several
  millennia ago can amount to several hours. The
  records of eclipses available from about 4000 BC
  is of considerable help to establish this
  variation of the length of the day.
• Seasonal variations due to meteorological causes,
  of periodic nature, and amplitude of few
  milliseconds
  
• Irregular fluctuations of geophysical origin,
  implying a transfer of angular momentum between
  core and mantle.
  

43
The irregularities of the day
Fluctuation of the duration of the day, expressed
as excess to 86400 SI, from 1995 to 1998 (adapted
from the IERS site, http//www.iers.org/).
44
The several realizations of UT
We can call UT0 the apparent one, determined by
the local Sidereal Time and longitude. It cannot
be used as is, in high precision works, because
of the polar motion, that can be removed by a
correction of the type
where (ux, uy) are the coordinates of the pole in
time units, and ?, ? the geocentric longitude
and latitude. Therefore UT1 is the
observatory-independent time, the one that should
enter in the previous formulae. It is determined
and distributed by the IERS. Although much more
uniform than UT0, UT1 is still affected by the
secular slowing down and by the non-uniformities
of the rotation (at the level of 1 part in 108),
and therefore not entirely satisfactory for
dynamical purposes. By removing from UT1 the per
iodic components, one derives UT2, which is
however not used in Astronomy.
45
The Time of the Ephemerides ET
To realize a truly uniform time, one could resort
to Newcombs Mean Sun, both as origin and rhythm.
The geometric mean longitude of the Sun is
T being expressed in Julian centuries after 1900,
Jan.0, 12h UT. This formula constitutes the for
mal definition of the Ephemerides Time ET, whose
rhythm is given by the coefficient of T, and
which has a small precessional acceleration. The
second of ET is defined by the number N of
seconds in the tropical year 1900
In other words, 1 second ET is the fraction 1/N
of the length of the tropical year 1900. At this
stage we should redefine the initial epoch as 12h
ET, not UT. A posteriori, it is seen that Newcom
b actually defined two different Mean Suns, one
whose Right Ascension increases uniformly with
UT, and one whose Right Ascension increases
uniformly with ET only if the rotation of the
Earth were strictly uniform the two would
coincide in a single body.
46
Weakness of ET
To free ET, whose origin is purely the annual
revolution, by the rotational slowing down of the
Earth, we must introduce a mobile Greenwich
meridian (ephemeris meridian), in very slow
motion toward East with respect to the
conventional one. The Hour Angle of ? with
respect to the ephemeris meridian is said
Ephemeris Sidereal Time. The two meridians were
assumed to coincide in 1902, at the present epoch
they are at about 2 from each other.
Furthermore, while UT can be obtained from
meridian observations, ET must be derived from
the longitude of the Sun the Sun however moves
too slowly along the ecliptic to provide a good
clock, the Moon is much better for this purpose.
So that ET can be better identified with the
argument (dynamical time) that enters in the
ephemerides of the Moon. But then, the knowledge
of ET implies the reduction of a great amount of
data, and the difference UT-ET is known only a
posteriori, and is affected by the residual
errors in the lunar ephemerides.
And finally, and perhaps more decisively, ET is
still a pre-relativistic concept therefore its
utilization in the Almanacs, introduced in 1960,
has been discontinued since 1984. Nevertheless,
it still retains some usefulness.
47
The International Atomic Time TAI
In the previous sections, the time has been
defined, or mathematically derived, by the motion
of heavenly bodies. Since 1955 a different
physical time of high regularity has become
available, namely the International Atomic Time
TAI, officially adopted in 1972. TAI is defined
by the radiation coming from two hyperfine levels
of the fundamental energy level of Cesium, when
the atom is far from magnetic fields and at sea
level. The frequency of this resonant transition
is 9 192 631 770 Hz, with a stability of about
2x10-13 (5 orders of magnitude better than UT1).
It defines the International Second SI, which,
always by definition, is equal to the second of
ET. The duration of the mean solar day is then 8
6400 SI. In practice, some 200 stations well
distributed over the Earth keep the Atomic Time
to within one nanosecond per day, and distribute
it via radio and navigational systems (e.g.
Loran-C, Omega, GPS).
48
The Terrestrial Dynamic Time TDT
Regarding the origin of TAI, by international
agreement its zero point was at epoch 1958 Jan 1,
at 0h UT1 (therefore at that date UT1 - TAI 0).
This decision implied an offset between ET and T
AI ET TAI 32s.184. Adopting now TAI
as the fundamental time scale, the quantity TAI
32s.184 is said Terrestrial Dynamic Time, TDT,
and since 1986 is the tabular argument of the
ephemerides TDT maintains the continuity with
ET, but its realization does not depend any
longer on observations of the Sun or of the Moon,
but on laboratory clocks. TAI (and so also TDT)
is certainly a very uniform time, nevertheless,
according to General Relativity, the frequency of
any clock varies with the varying gravitational
potential in which it is immersed. Therefore TAI
must be interpreted as a proper time. In the
differential equations of Mechanics it must be
transformed to coordinate time, according to the
position and velocity of the observer with
respect to the barycenter of the Solar System.
This ideal time of the inertial observer is said
Dynamical Time of the Barycenter (TDB).
49
The Time of the Barycentre TDB
The difference TDT-TDB is expressed by purely
periodic terms if a precision of a microsecond
is sufficient, the following expression can be
used
where E is the eccentric anomaly of the Sun, and
? , ? are latitude and longitude of the clock.
From 2001 onward, by IAU decision TDT has been r
enamed TT.
50
The Universal Coordinated Time UTC
At the end, however, what matters for Astronomy
(and also for navigation) is the true angle of
rotation of the Earth, namely UT in its various
realizations. Therefore, by international
agreement, a time is broadcasted having the
rhythm of TAI, but origin always coincident,
within 900 ms, with that of UT1. This hybrid time
is said UTC (Coordinated Universal Time). Because
UT is not uniform, UTC cannot be a continuous
function according to the need, a leap second is
added (or in theory subtracted, but since 1972
this never happened) at the beginning or at the
mid-point of each particular year. No need for
the leap second has been found since 1999 January
1st until further notice UTC-TAI -32 s (see
Bulletin C of the IERS). Although discontinuous,
UTC is therefore an extremely practical and
inexpensive time, and sufficient for many
astronomical purposes however, at least in
principle, it not correct to measure the duration
of an event by differencing UTC. The broadcasted
radio-signals actually contain also the
difference UTC-UT1.
51
The seasons - 1
We have affirmed that the tropical year
determines the succession of the seasons. At
epoch 1950.0 we had the following values
t0 1950 January 3.02
Seasons start when ?? 0 (spring), 90
(summer), 180 (autumn), 270 (winter).
Because no high precision is requested, in order
to find the corresponding values of M and of t,
we can ignore the term in sinM, obtaining  the
values of Table 4. 1, which contains also the
dates in leap years 2000 and 2096 (when the start
date will be the earliest of the 21st century)
52
The seasons - 2
The starting times delay by some 6h each year, in
a cycle of four years.

53
The seasons - 3
The two warm seasons last in the Northern
hemisphere 7 days longer than in the Southern
one on the other hand, the Earth is closer to
the Sun in the Southern summer. Averaged over the
globe, the sunlight falling on Earth at aphelion
is about 7 less intense than at perihelion
however, the average temperature of the whole
Earth at aphelion is about 2.3oC higher than it
is at perihelion this happens because there is
more land in the northern hemisphere and more
sea-water in the southern one. During the month
of July, the northern hemisphere is tilted toward
the Sun, and Earth's overall temperature
(averaged over both hemispheres) is slightly
higher, because the Sun is shining mostly on
continents, which have low heat capacity and heat
up more easily. January is the coolest month
because that's when our planet presents its
water-dominated, high heat-capacity, hemisphere
to the Sun. Southern summer in January is
therefore cooler than northern summer in July. In
order to satisfy the thermal balance of the Earth
as a whole, averaged over one year, efficient
mechanisms of heat transport are required, such
as regular winds and sea currents.
54
Historical records of climate
In this study, underground temperature
measurements were examined from 350 bore holes in
eastern North America, Central Europe, Southern
Africa and Australia. Using this unique approach,
Pollack et al. found that the 20th century to be
the warmest of the past five centuries.
http//www.ngdc.noaa.gov/paleo/globalwarming/paleo
last.html
55
The calendar - 1
The civil calendar adopted in many countries is
based on the length of the tropical year, because
the seasons follow the course of the Sun along
the ecliptic. However, this length cannot be
expressed by an integer number of days, not even
by a rational fraction. Several remedies were
adopted by the different cultures. In Rome, about
year 46 B.C. Julius Caesar agreed to the proposal
of the astronomer Sosigenes of adding one day to
the shortest month (February) each fourth year.
In the Julian calendar, the fourth year is called
bi-sextus, or leap year. In the first application
of this rule, some 90 days had to be suppressed
from the calendar. The extension of the Julian
calendar into the past (namely before its
adoption) is called proleptic calendar.
Because in Chronology year 0 does not exist,
passing directly from 1 B.C. to 1 A.D., in
performing calculations of intervals of time
between two events happened one before and one
after Christ, year 1 B.C. is year 0, year 2 B.C.
is year 1, and so on. Year 0 is considered a le
ap year.
56
The calendar - 2
With the Julian reform, the duration of the year,
averaged over a 4 year period, became of exactly
365.25 mean solar days, and the Julian Century of
36525 mean solar days. But this round number
fails the true duration by approximately 8 days
every 1000 years. After some further adjustment
made during the Council of Nicea (325 A.D.),
finally in 1582 Pope Gregorious XIII decreed to
suppress 10 days, jumping from Thursday October 4
directly to Friday October 15. As a further
element of the Gregorian reform, it was stated
that only the secular years divisible by 400
would be leap years. Therefore, following the
Gregorian reform, years 1600 and 2000 were leap
years, but not 1700, 1800 and 1900 in a cycle of
400 years there are only 97 of such leap years,
and the average duration over such cycle is
365.2425 mean solar days. Because in 400 years
there are 146097 days, which is evenly divisible
by 7, the Gregorian civil calendar exactly
repeats at each cycle of 400 years. However good,
the average value 365.2425 is still an
approximation to the true value, so that the
Gregorian calendar precedes the Sun by
approximately one day every 2500 years. As a
remedy, year 4000 could be considered a normal
one, not a leap one, but no agreement has been
reached.
57
The calendar - 3
Following Bessel, the year starts when the
longitude of the Fictitious Sun, affected by
aberration and referred to the mean equinox of
date, is exactly ?(F?) 280, and therefore
?(M?) 18h40m. Such instant, named epoch, is
always within 1 day from midnight of the Dec.
31st. In most applications, for instance in order
to calculate the amount of precession, this
slight difference between the start of the
Besselian year and of the civil year is entirely
negligible. The Besselian epoch is indicated by
the notation B1950.0 any other instant of time
during that year is indicated by the fraction of
year, e.g. B1950.45678. In order to refer the
Besselian year to the Julian calendar, it must be
recalled that the fundamental epoch B1900.0
corresponds to 1900 January 0d.813 1899
December 31, 19h31m (notice the astronomical
convention, year first, then month, day, hours,
and the utilization of the 0 for the last day of
the year). The calendar date of another epoch,
say B1950.0, is obtained by considering 50
tropical years since then, namely 18262.110 days,
or else 12.110 days more than 50 years of 365
days. Taking into account that 1900 was not a
leap year, subtract 12 days, add 0.110 to 0.813
to finally get B1950.0 1950 January 0d.923
1949 December 31, 22h09m. This is the civil date
of many stellar Catalogues such as the AGK3.
58
The calendar - 4

• The convention of Bessel remained valid until
  1984, when the IAU decreed to move the
  fundamental epoch to J2000.0 noon (not midnight!)
  2000 January 1d.5 UT (actually UT1), and to
  adopt Julian years of 365j.25 (or Julian
  centuries of 36525j).
• Therefore J1950.0 corresponds exactly to
  18262j.5 days before the fundamental epoch,
  namely to 1950 January 1d.0, differing by 1h51m
  from B1950.0.
• Two advantages have been achieved with that
  choice, namely
  
• the fixed duration
• the coincidence of the origin of the civil year
  with the Julian Day.

59
The Julian Day (JD)
In astronomy, it is customary to count the
passage of time in Mean Solar Days starting from
an initial arbitrary date. For historical
reasons, such initial date is the mid-day (not
midnight!) of Jan. 1st 4713 B.C. (in Chronology,
year 0 doesnt exist, therefore 4713 B.C. -
4712). Such system of dates is expressed in
Julian Days (JD). Thus, 1950 Jan. 1st, 12h UT,
corresponds to JD 2433283.0, and similarly
B1950.0 JD 2433282.423 , J2000.0 JD 24515
45.0   The inverse is Julian epoch J J2000.0
(JD - 2451545)/365.25   Besselian epoch B B
1900.0 (JD - 2415020.31352)/365.2422
  In order to calculate how many days separate tw
o date, the correct procedure is to calculate the
two corresponding JDs, and then to make the
difference.
60
The Modified Julian Day (MJD)
In order to avoid carrying too many decimals, and
to start the day at midnight, a Modified Julian
Day (MJD) has been introduced, having its zero
date on 1858 Nov. 17.0   MJD JD - 2400000.5
The JD scale furnishes a continuous reference of
time however, this scale is as uniform as the
mean solar day itself. But the duration of the
day has a secular decrease, so that JD is not
entirely satisfactory for dynamical purposes over
intervals of centuries or millennia.
The Explanatory Supplement calls Julian Date wha
t we call here Julian Day, and reserves the name
of Julian Day to its integer part we prefer
Julian Day, in order not to confuse it with a
date in the Julian calendar.
61
Esercizi svolti - 1
Si supponga che la posizione di Giove sia
(11h40m30s.4, 944'39"), e che quella del suo
sesto satellite sia (11h38m05s.4, 1010'57"). Si
calcolino l'angolo di posizione e la distanza
relativa a Giove del satellite.
Trasformiamo tutti i dati numerici in gradi e
decimali Giove 11h40m30s.4 11h.6751111 175.
1266667 , 944'39" 9.7441667
Sesto satellite 11h38m05s.4 11h.6348333
174.5225000 , 1010'57" 10.1825000
Siccome i due corpi sono molto vicini, basterà
calcolare la distanza angolare come se il
triangolo sferico definito dai due corpi e
dallintersezione del parallelo per la sesta Luna
e il cerchio orario di Giove fosse piano
62
Esercizi svolti - 2
In altre parole, si deve fare attenzione che
larco del parallelo per la sesta Luna e il
cerchio orario di Giove viene accorciato del cos?
, e che per precisione lievemente migliore si è
preso come ? il valor medio tra le ? dei due
corpi. Infine, il sesto satellite ha ? minore di
quella di Giove ma ? maggiore.
Quindi rispetto alla figura, X2 è Giove e X1 è
la sesta luna. Ma lesercizio chiede langolo di
posizione rispetto a Giove. Tenendo conto che
langolo di posizione parte dal Nord verso Est,
si capisce che la luna si situa nel quarto
quadrante, per cui langolo di posizione si può
calcolare semplicemente come
p 270 35.948 ? 306.40
63
Esercizi svolti - 3
Calcolare il JD corrispondente all8 luglio1993,
6h UT
seguiamo un metodo laborioso e non adatto a una
programmazione generale, ma che è illustrativo.
All'1 gennaio 1993, 12h UT mancano 7 anni al
J2000.0, cioe 7x365 1 giorni (dato che il 1996
e anno bisestile) 2556 giorni, e pertanto si
ha a quella data JD 2448989.0 (sempre alle 12h
UT). Per calcolare ora il JD all8 luglio
conviene, secondo luso astronomico, riferirsi
allo 0 gennaio, in modo da contare i giorni
progressivi con il loro numero di calendario. L'8
luglio corrisponde pertanto a altri 189 giorni.
Da cui   0 gennaio 1993, 0h UT JD 2448987.5,
0 gennaio 1993, 12h UT JD 2448988.0
  1 gennaio 1993, 0h UT JD 2448988.5, 1 gennaio
1993, 12h UT JD 2448989.0   8 luglio 1993,
0h UT JD 2449176.5, 8 luglio1993, 12h UT JD
2449177.0 Sempre con questo metodo, si rifaccia i
l calcolo per il 13 febbraio del 1993, partendo
pero' dal 1900 e sapendo che alle 0h UT dello 0
gennaio 1900 si aveva JD 2415019.5.
64
Esercizi svolti - 4
Si svolga ora l'esercizio inverso, dato JD
2449238.5 ricavare la data del calendario
vediamo il metodo più semplice (ma non adatto a
un programma di calcolo generale), partendo dal
1900.0 Sia JD 2449238.5, e poniamo NY JD - J
D(0 gennaio 1900 0h UT) 34219.
Il numero di anni passati dal 1900 e'
int(NY/365.25) 93, quindi siamo nell'anno Y
1993. Il numero di anni bisestili e' stato dunqu
e pari a int(93/4) 23. Il numero di giorni pas
sati dall'inizio dell'anno 1993 e'
  ND ?(NY? (93?365)-23? 251 dallo 0 gen 1993
.   siamo quindi al 251-mo giorno di un anno che
non è bisestile, e che cade quindi in settembre.
Dato che all'1 settembre sono trascorsi 243
giorni dallo 0 gennaio, si ha finalmente che
siamo all'8 settembre 1993, 0h UT.
65
Esercizi svolti - 5
Calcolare il TS medio a Greenwich, per un giorno
e ora qualunque dell'anno 1993.
  Dalle Lezioni (formula 10.1) si ha, trascurando
termini in T2 e T3 TSMG (0hUT ) 6h41m50s.54
8 8640184s.8129T 6h.697375 240h.051337T
  essendo T in secoli giuliani dal J2000.0. Ma
lo 0 gennaio 1993, 0hUT si ha JD 2448987.5, da
cui  
TSMG ( 0 gen 93 0h UT ) 6h.6444990
66
Esercizi svolti - 6
e per una data qualunque, essendo d il numero di
giorni trascorsi dallinizio convenzionale
dellanno (il 1 gennaio d 1, il 1 febbraio d
32, etc.) a unora t qualunque
  TSMG ( d , t h UT ) 6h.6444990 0h.0657098 d
1.002737909 t h   Per es., l8 luglio ha d
189 (lanno non è bisestile), per cui TSMG (189,
0h UT ) 19h.0636512 19h03m49s.160.
Alle 9h44m UT si ha TSMG (189, 9h44m UT)
19h.0636512 1.00273790x9.733 4h49m25s.081
  2) calcolare alla stessa data il TS medio in un
a località di longitudine ? (espressa in ore
minuti e secondi). Si prenda il risultato ottenut
o in 1) e si aggiunga la longitudine (se a Est) o
la si sottragga (se a Ovest), ottenendo cosi il
TS locale medio. Ad es. per il TNG si tolga 1h11
m33s.37
67
Esercizi svolti - 7
Determinare il TS a Greenwich nell'istante in
cui a Asiago (telescopio di 122 cm) il tempo
siderale locale è TS 4h38m47s.26
Per ricavare il TS a Greenwich abbiamo bisogno
della longitudine del telescopio di 122cm, che
è ? 113140.73 11.527981 0h.7685320
0h46m06s.72. Siccome Asiago è a Est di Greenwich
avremo   TSG TSA - 0h.7685320 4h.6464
610 - 0h.7685320 3h.8779290 3h52m40s.54
68
Esercizi svolti - 8
Si vuole ora, nello stesso esercizio, il TS
apparente Si deve aggiungere lequazione delleq
uinozio EE che possiamo calcolare, con precisione
modesta ma sufficiente ai nostri scopi con il
seguente procedimento lespressione di EE e
(Lezioni, Cap.10)
di cui qui consideriamo il solo primo termine che
dipende dalla longitudine del nodo ascendente
della Luna sulleclittica (formula 5.8)
essendo t in anni giuliani a partire dal J2000.0.
Quindi ci limitiamo a calcolare le espressioni
Allo 0 gennaio1993, 0h UT, t -7 , 260?.44, EE
1s.03 all8 luglio 1993, 0h UT , t -6.5178,
?N 251?.12, EE 0s.995 I valori precisi rip
ortati nellAstronomical Almanac sono piu
piccoli di questi per circa 0s.05.
69
Esercizi da svolgere
1 - Le coordinate equatoriali del Sole siano AR
6h40m.2, Dec 23.1. Che giorno dell'anno è?
Quale sarà la massima altezza del Sole
sull'orizzonte di Asiago? Quali saranno le
coordinate eclitticali approssimative?
2 Calcolare il diametro apparente del Sole in
funzione della data 3 - Esprimere algebricament
e le seguenti date 31 gennaio dell'anno 1 avant
i Cristo 31 gennaio dell'anno 1 dopo Cristo e
i corrispondenti JD. Calcolare poi quanti giorn
i sono trascorsi tra le due date.