07 02 05C. Barbieri Elementi_AA_2004- 05_Quarta settimana 1 Lezioni IV settimana L'aberrazione della...

07 02 05 C. Barbieri Elementi_AA_2004-05_Quarta settimana

1

Lezioni IV settimana

L'aberrazione della luce

La deflessione gravitazionale della luce

Il tempo in astronomia

Stagioni, calendario

Esercizi

Anche questa parte è in inglese!


2

The aberration of lightAs seen in the previous chapters, precession and nutation are phenomena due to the variable orientation of the observer’s 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:

V 2 30 km/sa

naP

being P the sidereal year, and n ( 3548”/day) the so called mean motion.The light will cross the 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:

V2 0.0001a

aK n

Pc c (radians), or: K 20”.6


4

The solar aberration - 2

2V

1r

na

e

2

1V V

601t r

nae

e

e 0.0167 is the small ellipticity

a is the semi-major axis

The point P is the perihelion.

The Earth's velocity vector V can be decomposed in a component perpendicular to the radius vector, and one perpendicular to the semimajor axis:

We can easily derive a more accurate description by taking into account the elliptical shape of the orbit:

Notice that Vt is constant not only in amplitude but also in direction!


5

The solar aberration - 3

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 ecliptic longitude between the aberrated (observable) and the geometric (unobservable) Sun is:

2

V20".495

1r na

Kc c e

The constant of the solar aberration is then more properly:

1 cos( )K e € €

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 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 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.


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. Bradley’s discovery conclusively proved the correctness of Roemer’s hypothesis, gave a direct way to determine K and a second, and much more precise, way to determine c.


8

The annual aberration - 3An 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 Earth’s 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 Vt = ECV/c. 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:

V Vsin( ') sin sin ' sin( )

c c

being the component of the Earth velocity perpendicular to the apparent direction of the star.

Vsin '

c


9

The annual aberration - 4Notice 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 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 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 aberrationThe 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:

2

rel Gal

1 Vsin sin sin 2 '

2 c

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:

1. neglect of higher order terms in trigonometric expansions2. 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 Earth’s 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 SS’U; after simple passages, one gets:

( ' ) cos cos cos( )K € ( ' ) sin ( )sin K €

During the course of the year, the star traces the locus:

2

2 2cossin

K

which is the annual aberrational ellipse, whose semi-major axis is K.


13

Effect of the ellipticity of the orbitLet 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:

2

V20".495

1r na

Kc c e

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):

coscoseK sin sin eK


14

Annual Aberration in equatorial coordinates -1

sin sin cos cos cos 1 sin cos'

cos cos' (sin cos cos cos sin sin cos sin sin cos )

1 ( cos sin sin sin cos )

X YK

cK

X Y Zc

€ €

€ € €

By ignoring the E-terms, and calling

),,( ZYX

the equatorial components of the Earth's velocity vector, whose approximate values are:

0.0172 sin X €

0.0158 cosY € 0.0068 cosZ €

(in AU/day, c = 173.14 AU/day) we have:


15

Annual Aberration in equatorial coordinates -2In order to remove the elliptical aberration, we must add the (almost) constant terms:

sin sin cos cos cos / cosKe

sin cos cos cos sin sin cos sin cosKe

for the FK5 and all catalogues based on it. The corrections therefore take the form:

DdCc ' ''' DdCc in which C, D depend on the sun’s 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 aberrationThe 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 Earth’s center, the difference is:

1510292.7 s

0,6LST h

d -d 0 .021( cos '/ ) cos / cossHA c HA

d 0".320( cos '/ )sin senc HA


17

Stellar and planetary aberrationWe 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 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. 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 - 1There 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 Einstein’s 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 Schwarzschild’s radius S given by:

22S

GM

c

whose value is about 3.0 km for the Sun and 0.88 cm for the Earth.


19

The gravitational deflection of the light - 2The 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.Q QR a

€ €

The constant Q is equal to 2 in the Newtonian theory, and to 4 in General Relativity; therefore is 0”875 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%

,a € €


20

The gravitational deflection of the light - 3The 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:

2 1 cos E 1E 0".00407

EsinE tan2

a

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 AstronomyIn 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 associated 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 rotationThe 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 Earth’s figure.

Which is the duration of the diurnal 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 timeWe 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:

sin cot tan € €

In other words, ST is defined by the Sun, not by the stars.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).

Notice, in the spherical triangle one side might be larger than 180°


24

The non-uniformity of the Sidereal TimeHowever, 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.

cosEE


25

The annual revolutionThe 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 Kepler’s 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:

0

1 11 cos( )e

r p a

p

e

1 2 b a e 1 2

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:

2d 1 d

d 2 d 2

A CA r

t t

where r is the instantaneous distance Earth-Sun.


26

The Solar Time - 1Let 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 geometrical point does not belong to the equator, but to the ecliptic, moving on it according to Kepler’s first two laws; those two factors affect both the duration and the uniformity of the solar time.


27

The Solar Time - 2Indeed, 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-uniformity of T⊙ is the same of that of ⊙

(disregarding in the present context the minute accelerations of ).Let ⊙, ⊙, ⊙ be respectively the ecliptic longitude, right ascension and

declination of the Sun; the following relations can be easily derived:

sin sin sin € € tan tan cos € €

Taking the time derivative of the second and inserting the first we also get:

2 2 2

cos cos

1 sin sin cos

€ € €

€ €

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

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.

To quantify the non-uniformity of we must take into account that: Kepler’s 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:

-161'.1 4 4 jm s € around the second of January

-157 '.2 3 49 jm s €

around the 2nd of July

11 j €


30

The non-uniformity of the Solar Time - 3Therefore 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 we’ll 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 same daily motion, which has the value:

" 3548".3n

a value which derives from the length of the tropical year (period of time between

two consecutive passages of the Sun through point ).

per day


31

The equation of the centre - 1The difference: (F )EC € €

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:

( ) ( ) ( ) ( ) ( ) 2 sin ( ) ( ) ( ) 2 sin ( )t t M t e M t M t e M t €

( ) (F ) 2 sin ( ) 0.03345 'sin ( ) 115'sin ( )EC t e M t R M t M t € €


32

The equation of the centre - 2The 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:

only in two occasions, when the Sun passes through the semi-minor axes of its orbit.

0 €

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:

-1(1 2 cos ) 3548".3 118".7cos jn e M M €


33

The Equation of Time - 1

2cos 1 1 cos 1

sin 2 sin 4 cos 1 2 cos 1

148'.1sin 2 3'.2sin 4

€ € € €

€ € €

2

s s0 0 0

( ) ( ) 2 sin tan sin 2( ( ) )2

460 .3sin ( ) 592 .2sin 2( )

M e M M

A nt n t t A nt

€

From the relation

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: (F ) = (M ) € €

tan tan cos € €

which is a form of the already discussed transcendental equation, we derive:


34

The Equation of Time - 2Finally, calculate the equation of time E, namely the difference:

s0( ) 460 .3sin ( ) 592 .2sin 2( ( ) )sE M n t t M € €

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 UTFor 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⊙):

hT(M ) (M )+12HA€ €

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 rhythmsBy definition therefore, the sidereal time ST and the Mean Solar Time T(M⊙) have

the same degree of uniformity of the Earth’s 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 = 365j.2421988 = 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⊙)


37

Relation between UT and ST - 1h

(Greenwich)UT = (M ) +12HA €

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. According to Newcomb, the mean longitude of the non-aberrated F⊙ at 12h UT (noon) of Jan. 1st, 1900

had the value:

(F⊙) = 28040’56”.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.

By definition then:


38

Relation between UT and ST - 2After 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.812866T + 0s.093104T 2 -

6s.210-6T 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 we will discuss later.


39

The year - 1The 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.00000614T; 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'/jwhose 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 Earth’s 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 between 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 from Kepler’s 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 timeWe 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 UTWe 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:

'tan)cossin(0UT1UT yx uu

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 periodic components, one derives UT2, which is however not used in Astronomy.


45

The Time of the Ephemerides ETTo realize a truly uniform time, one could resort to Newcomb’s Mean Sun, both as origin and rhythm. The geometric mean longitude of the Sun is:

2089".113".12960276804".48'41279 TT

T being expressed in Julian centuries after 1900, Jan.0, 12h UT. This formula constitutes the formal 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:

9747.3155692513.129602768

86400355251296000

N

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 Newcomb 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 ETTo 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 TAIIn 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 86400 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 TDTRegarding 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 TAI:

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 TDBThe difference TDT-TDB is expressed by purely periodic terms; if a precision of a microsecond is sufficient, the following expression can be used:

3 6

TDB TDT

1.658 10 (sin 0.0368) 2.03 10 cos '(sin(UT ) sin )E

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 renamed 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 - 1We have affirmed that the tropical year determines the succession of the seasons. At epoch 1950.0 we had the following values:

282 '04 '30" 115'sinM M €

03548".3( )M t t 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 - 2season Start (1950) Duration

(days)Start (2000) Start (2096)

spring 21.2 March 92.81 20.3 March 19.5 March

summer 22.0 June 93.62 21.0 June 20.1 June

autumn 23.1 Sept. 89.82 22.7 Sept. 21.9 Sept.

winter 22.4 Dec. 89.00 21.5 Dec. 20.9 Dec.

The starting times delay by some 6h each year, in a cycle of four years.


53

The seasons - 3The 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

http://www.ngdc.noaa.gov/paleo/globalwarming/paleolast.html

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.


55

The calendar - 1The 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 leap year.


56

The calendar - 2With 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 - 3Following 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:

1. the fixed duration

2. 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 doesn’t 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 2451545.0 The inverse is:

Julian epoch J = J2000.0 + (JD - 2451545)/365.25

Besselian epoch B = B1900.0 + (JD - 2415020.31352)/365.2422 In order to calculate how many days separate two 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 what 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 - 1Si supponga che la posizione di Giove sia (11h40m30s.4, +9°44'39"), e che quella del suo sesto satellite sia (11h38m05s.4, +10°10'57"). 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 , +9°44'39" = 9°.7441667Sesto satellite: 11h38m05s.4 = 11h.6348333 = 174°.5225000 , +10°10'57" = 10°.1825000

Siccome i due corpi sono molto vicini, basterà calcolare la distanza angolare come se il triangolo sferico definito dai due corpi e dall’intersezione del parallelo per la sesta Luna e il cerchio orario di Giove fosse piano:

2 2 2 2 2cos 0.604 0.97 0.438 0 .739 44 '.312

G Sd


62

Esercizi svolti - 2In altre parole, si deve fare attenzione che l’arco 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 l’esercizio chiede l’angolo di posizione rispetto a Giove. Tenendo conto che l’angolo di posizione parte dal Nord verso Est, si capisce che la luna si situa nel quarto quadrante, per cui l’angolo di posizione si può calcolare semplicemente come:

0.438tan 0.7252

0.604p

p = 270° +35°.948 306°.40


63

Esercizi svolti - 3Calcolare il JD corrispondente all’8 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 all’8 luglio conviene, secondo l’uso 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 2449177.0Sempre con questo metodo, si rifaccia il 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 - 4Si 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 - JD(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 dunque pari a int(93/4) = 23. Il numero di giorni passati dall'inizio dell'anno 1993 e': ND = (NY (93365)-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 - 5Calcolare 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.548 + 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

070205.036525

0.24515455.2448987

T


66

Esercizi svolti - 6e per una data qualunque, essendo d il numero di giorni trascorsi dall’inizio convenzionale dell’anno (il 1 gennaio d = 1, il 1 febbraio d = 32, etc.) a un’ora t qualunque:

TSMG ( d , t h UT ) = 6h.6444990 + 0h.0657098 d + 1.002737909 t h

Per es., l’8 luglio ha d = 189 (l’anno 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 una località di longitudine (espressa in ore minuti e secondi).Si prenda il risultato ottenuto 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 1h11m33s.37


67

Esercizi svolti - 7Determinare 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: = +11°31’40”.73 = 11°.527981 = 0h.7685320 = 0h46m06s.72.Siccome Asiago è a Est di Greenwich avremo:

TSG = TSA - 0h.7685320

= 4h.6464610 - 0h.7685320 = 3h.8779290 = 3h52m40s.54


68

Esercizi svolti - 8Si vuole ora, nello stesso esercizio, il TS apparente Si deve aggiungere l’equazione dell’equinozio EE che possiamo calcolare, con precisione modesta ma sufficiente ai nostri scopi con il seguente procedimento: l’espressione di EE e’ (Lezioni, Cap.10):

di cui qui consideriamo il solo primo termine che dipende dalla longitudine del nodo ascendente della Luna sull’eclittica (formula 5.8):

cosEE

essendo t in anni giuliani a partire dal J2000.0. Quindi ci limitiamo a calcolare le espressioni:

2125 .04452 19 .3413626 0 .00002971N t t

N sen2".17 NEE sencos2".17

Allo 0 gennaio1993, 0h UT, t = -7 , 260.44, EE = + 1s.03all’8 luglio 1993, 0h UT , t = -6.5178, N = 251.12, EE = +0s.995 I valori precisi riportati nell’Astronomical 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 algebricamente le seguenti date:31 gennaio dell'anno 1 avanti Cristo31 gennaio dell'anno 1 dopo Cristoe i corrispondenti JD. Calcolare poi quanti giorni sono trascorsi tra le due date.

07 02 05C. Barbieri Elementi_AA_2004- 05_Quarta settimana 1 Lezioni IV settimana L'aberrazione della...

Documents

Transcript of 07 02 05C. Barbieri Elementi_AA_2004- 05_Quarta settimana 1 Lezioni IV settimana L'aberrazione della...