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

1
Elementi di Astronomia e Astrofisica per il Corso
di Ingegneria AerospazialeParte III
Parallasse Diurna e Annua Velocità radiali e Moti
propri Esercizi
Avvertenza questa parte è in larga misura
opzionale, ed è stata svolta molto parzialmente a
lezione. Basta leggerla e ritenere gli elementi
fondamentali effettivamente svolti.
2
La Parallasse
The phenomenon of the parallax is due to the
finite distance of the star from the observer.
Two observers located in two different positions,
or the same observer moving from one place to
another because of the diurnal rotation or annual
revolution, will see an object in two different
locations on the celestial sphere. Of all the
phenomena described so far, that alter the
apparent direction of the source, the parallax is
the first to give information on the distance,
and therefore on the nature, of the heavenly
body, and not only on the motions of the observer
or on fundamental properties of light.
Furthermore, the knowledge in meters of the
distance between the observers provides a link
between the terrestrial and the cosmic distance
scales. The determination of the parallaxes is
therefore one of the most fundamental
astronomical measurements.
3
La parallasse trigonometrica -1
Given two observers, O, C at a distance R from
each other, and an object X at distance d from
O, and d from C, observer O will see the object
in direction z, and observer C in direction z
z p in respect to the baseline OC
Fg. 8.1 - The geometric parallax. The angle p is
called the parallax of X in respect to the
baseline OC
4
La parallasse trigonometrica -2
The following exact relations hold
5
La parallasse trigonometrica -3
These generic relationships, useful in
terrestrial triangulations, or when observing an
artificial satellite, will now be applied to the
astronomical case. In the following well
distinguish between objects in the solar system,
where the Earths radius is sufficiently large to
provide an useful baseline (diurnal parallax),
and stars for which this radius is exceedingly
small, and the baseline is provided by the radius
of the annual revolution around the Sun (annual
parallax).
6
La parallasse diurna - 1
When determining the celestial coordinates of an
object of the solar system (say a planet, but the
same will hold for a comet or an asteroid), it
will be necessary to take into account the
location of the observer on the terrestrial
surface, namely his topography, in order to
translate these coordinates to an ideal
geocentric observer. This process will therefore
transform topocentric into geocentric
coordinates. Be now C the center of the Earth, O
the generic observer on the surface, at a
distance from C (? ? 1), X the planet distant d
from C and d from O (see Figure), the angle is
the instantaneous parallax of X.
The instantaneous diurnal parallax. The
ellipticity of the Earth is greatly exaggerated.
The plane COC doesnt necessarily coincide with
the meridian of C.
7
La parallasse diurna - 2
In the Figure line OZ is the astronomical
vertical, line CZ the geocentric vertical the
observations provide the astronomical zenith
distance z0, than can be transformed into the
distance from the geocentric vertical z by means
of the deviation of the vertical v ?-? . The
angle p is then given by
The angle p is variable with the rotation of the
Earth, even disregarding the motion of the
planet. It has been agreed therefore to name
horizontal diurnal parallax ? the angle under
which X sees perpendicularly the Earth radius. In
other words, ? is the maximum value of p at any
given distance of X from C, but ? itself will
vary according to the relative positions of X and
C in their heliocentric orbits.
8
Parallassi diurne -3
If the observations provide p , we can derive d
and z from these equations. Notice that the
topocentric zenith distance is larger than the
geocentric one, and that the change in line of
sight form the topocentric to the geocentric
observer takes place in the plane OCX. This plane
essentially coincides with the vertical plane of
X as seen by O (disregarding the small difference
between astronomical and geodetic vertical), and
therefore the influence of the diurnal parallax
is practically all in zenith distance (or in
elevation above the horizon).
9
La parallasse diurna - 4
The horizontal parallax of the Moon varies
between 54 and 61 due to the strong
eccentricity of its geocentric orbit (notice that
these values are about twice the apparent
diameter of the lunar disk, therefore the
occurrence of a lunar occultation depends very
critically on the position of the observer). All
other celestial bodies are much farther than the
Moon, apart some occasional object than can pass
inside its orbit, and their diurnal parallax can
vary by great amounts. For instance, ? (Venus)
varies between 5 and 34. Therefore, essentially
in all cases (except artificial satellites) we
are justified to assume
and ? very small, so that
10
Parallassi diurne -5
All other celestial bodies are much farther than
the Moon, apart some occasional object than can
pass inside its orbit, and their diurnal parallax
can vary by great amounts. For instance, ?(Venus)
varies between 5 and 34. Therefore, essentially
in all cases (except artificial satellites) we
are justified to assume
and ? very small, so that
,
For the Sun, averaging over the slight
ellipticity of the orbit, the distance d is the
astronomical unit 1 AU ? 1.49x108 km,
the horizontal parallax is essentially constant,
so that
11
La parallasse diurna del Sole
For the Sun, averaging over the slight
ellipticity of the orbit, the distance d is the
astronomical unit
1 AU ? 1.49x108 km, the horizontal parallax is
essentially constant, as will be discussed in
greater detail, so that
Expressing the distance d to a generic body in AU
and the parallax in seconds of arc
12
La parallasse annua - 1
Annual parallaxes are the fundamental method for
the direct determination of the stellar
distances.
Fig. 8. 4 The annual parallax.
Fig. 8. 4 shows the Earth in two diametrically
opposed positions along its orbit, which at
moment we take as circular with radius 1 AU. Be
E1 and E2 the two geocentric observers, S the
ideal heliocentric observer, and X a nearby star,
which will be seen from the Earth as projected in
X1 and X2 on the celestial sphere (in other
words, with respect to the background of the
distant stars).
13
La parallasse annua - 2
Calling ES the direction Earth-Sun, z the angle
of the line of sight with that direction from S
and z from E, d the heliocentric distance of X,
we have
The angle ?, under which the Astronomical Unit is
seen perpendicularly from the star, namely the
maximum value of p, is called the annual parallax
of X. Till now, star Proxima Centauri (in the
triple system of ? Cen) has the largest observed
value, ? 0.76 no sensible error is therefore
made in using arcs instead of their sine or
tangent, and therefore
the geocentric observer sees the star closer to
the Sun by the slight amount (z-z) on the
celestial sphere this apparent movement occurs
along the great circle passing through X and S.
14
La parallasse annua - 3
During the year, S moves along the ecliptic, and
the locus X of apparent geocentric positions
will be an ellipse centered on the heliocentric
position X. Notice that this ellipse has nothing
to due with the ellipticity of the Earths orbit
(that we have assumed circular), it is simply the
projection effect depending on the ecliptic
latitude ? of X the slight ellipticity of the
orbit (e 0.0167) will introduce small
modifications, that here we neglect.
Fig. 8. 5 The annual parallax in ecliptic
coordinates   If (?, ?) are the heliocentric
coordinates of X, the geocentric coordinates will
differ by a very small amounts that we can treat
as differentials, (???, ???), in other words
the small triangle XXX can be considered as a
plane triangle with infinitesimal sides
15
La parallasse annua in coordinate eclittiche
Calling ?? the longitude of the Sun, after some
calculations (similar to those made for the
annual aberration), we get
16
Il parsec
Using the Astronomical Unit as baseline, it is
possible to institute the fundamental unit of
astronomical distances, namely the parsec one
parsec (pc) is the distance from where the AU
subtends perpendicularly an angle of 1.
Therefore   1 pc 206264.8 AU 3.09 x1013
km   where the last conversion factor is derived
from the solar parallax. Any revision of this
latter value will not change the distance as
expressed in parsecs. On the practical side, this
remark is not so important today, because the
precision with which the solar parallax is known
is much higher than the precision of stellar
distances it is useful to remember it, however,
considering how the cosmic distances ladder has
been connected to the laboratory units. A
secondary unit of distance is the light-year
(l-y), corresponding to the distance traveled in
vacuum by the light in one year it is easily
found that 1 pc ? 3.26 l-y.
17
Parallassi secolari
The Solar System moves in respect to the ensemble
of the nearby stars with a velocity of some 20
km/s in direction of the constellation Lyra (a
point named apex of solar motion). An hypothetic
observer, at rest in the frame of this group of
nearby stars, would observe the Sun in
rectilinear motion toward this apex, and the
planets describing open orbits around it the
distance covered by the traveling Sun is of
approximately 4 AU per year. This baseline is
four times that of the annual parallaxes, and we
call secular parallax H of the star X the angle
under which this baseline is seen perpendicular.
Lets call (km/s) the velocity of the Sun, n the
number of seconds in one year, d (km) the
heliocentric distance of X after simple
conversion of the relevant units we find
However, this is a secular, not a periodic,
motion, and it is observed entangled with the
proper motion of that particular star. It is
however an useful distance indicator for a group
of stars having the same distance, as well
discuss also in the next lectures.
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Parallassi dinamiche - 1

• Caso del sistema solare
• consider Keplers third law (see also chapter 12)
  for a given planet, in its
• approximate expression

being P the sidereal period, a the semimajor axis
and M? is the mass of the Sun. Comparing this
with the Earth
we can derive only the relative dimensions of
each orbit. At least one absolute determination
in km is needed to fix the scale of the Solar
System. To reach this goal, it is advantageous to
observe from several locations an asteroid
(having a star-like image) whose orbit takes it
as close as possible to the Earth. Several
asteroids were tried, the one giving the best
results being 433 Eros, whose orbit is partly
internal to that of the Earth (Eros was reached
in 2001 by the spacecraft NEAR, who crashed on
its surface at the end of a very successful
mission).
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Parallassi dinamiche - 2
Eros came at the opposition in 1900-1901, and
again in 1930-1931. The first event, during which
the minimum distance was 0.32 AU, was observed
visually and gave the value
In the second opposition, the small planet came
much closer, 0.17 AU, and the observations took
advantage of photography the derived value was
Eros was observed again in 1975, but this time
with radar echoes, a technique that had been
successfully applied in 1959 and 1961 to the
inferior transits of Venus these radar data gave
also the first determination of its dimensions,
approximately 16x35 km.
20
Parallassi dinamiche - 3
b) Coppia di stelle be A, B the two components of
a binary star, having respectively MA and MB
masses, and orbital period P. Keplers third law
states that
In astronomical units (M in solar masses, P in
years)
(pay attention to ? 3.1415... in the first
formula, and ? parallax in the second). In the
second equation, a and ? are both in arcsec.
Masses in general are unknown, but putting M 2
an acceptable value for ? is obtained, because of
the small weight of M on the error. The method
can be refined by using an appropriate
mass-luminosity function, calibrated on well
known systems, but usually the uncertainties on a
and P dominate the error on ?.
21
Velocità radiali -1
From a formal point of view, if r is the
heliocentric (or better barycentric) position
vector of a star at a given initial epoch t0, and
its distance in terms of the trigonometric
parallax ?, the velocity vector could be easily
derived
where ??0, ??0 are the proper motions in Right
Ascension and declination, and is the radial
velocity. The position of the star at a following
date t1, but referred to the same epoch, would
then be derived from
22
Velocità radiali -2
However, the knowledge of the needed quantities
is usually incomplete, so that in the following
the radial velocities and the proper motions will
be considered separately. It also clear that the
treatment of the system of proper motions and
velocities is fundamentally linked to that of
precession, a great complication indeed for
transferring catalogues based on stellar
observations from one epoch to another, if the
highest precision is to be maintained. The
radial velocity is measured through the Doppler
effect, namely through the variation in
wavelength ? of the radiation, caused by the
relative motion Vr along the line of sight the
velocities of the planets or of the stars of the
Milky Way are usually so small in comparison with
the velocity of the light c that no sensible
error is made in using the pre-relativistic
formula
where ?O is the wavelength measured by the
observer, and ?S is that measured in the
rest-frame of the source. The use of letter z is
fairly widespread.
23
Velocità radiali -3
Notice that the radial velocity can be positive
(z gt 0, namely the wavelength is red-shifted), or
negative (z lt 0, namely the wavelength is
blue-shifted). When the velocities exceed say
0.01c, then Special Relativity must be taken into
account. It is more advantageous here to work in
terms of frequency ? than of wavelength ?. In
vacuum
the classical formula would be
In the relativistic formula
where n is the normal to the wavefront. It is
seen therefore that Special Relativity foresees a
transverse Doppler effect (when n is
perpendicular to V), which is not present in the
pre-relativistic formula. In other words, the
whole velocity vector, and not only its radial
component, enter in the observed frequency
displacement.
24
Velocità radiali - 4
After simple passages, we derive
?V/c, and angle ? ?-? is the angle between
the line of sight and the velocity vector of S.
If the velocity is all radial
25
Velocità radiali nel Sistema Solare

• Solar System velocities of few tens of km/s
  prevail, with the notable exception of comets and
  asteroids skimming the surface of the Sun, whose
  heliocentric velocity can exceed 700 km/s. The
  classical formula is therefore adequate for most
  applications.
• However, another factor must be carefully taken
  into account, that of the extreme precision (say
  ?1 mm/s) with which the velocity of a spacecraft,
  possibly orbiting a planet, can be measured. The
  consequence is that for accurate navigation
  inside the Solar System, General Relativity must
  be taken into account in expressing the metric of
  space-time.

26
Velocità radiali delle stelle
b) The radial velocities of the normal stars of
the Galaxy rarely reach 500 km/s among the
nearest stars, velocities higher than 50 km/s are
seldom encountered, with notable exceptions such
as Barnards star, which moves at 108 km/s in
respect to the Sun. For the great majority of
stars, a precision of at best ?100 m/s is
reached only in favorable cases and with refined
techniques, e.g. in searches for extra-solar
planets by means of radial velocity variations,
?3 m/s are achieved, the limiting factors being
on one side the technical limitations and on the
other the turbulent structure of the stellar
spectral lines themselves. Therefore the
classical formula is usually adequate. There are
cases however where Special Relativity must be
applied, e.g. for the variable star SS433, or for
the expanding gaseous envelopes of explosive
variables (novae, supernovae), where velocities
of tens of thousands km/s are encountered.
27
Velocità radiali delle galassie
c) For galaxies, beyond a certain distance
roughly coinciding with that of the cluster of
galaxies in Virgo, whose velocity is of about
1000 km/s, only positive velocities are
encountered (for closer galaxies, negative
velocities can be found, e.g. for M31 in
Andromeda). The spectral lines of the distant
galaxies and of other objects of cosmological
significance, e.g. the Quasi Stellar Objects
(named also Quasars, or QSOs), are always
red-shifted, with an amount zc increasing with
the distance, as was discovered by E. Hubble
around 1930. This observational effect is at the
basis of all Cosmology, and is referred to as
expansion of the Universe. Values of zc up to 6
have already been measured (so that for instance
the spectral line of Hydrogen Lyman-?, with ?
1216 Å as measured in the laboratory, is observed
at ? 8512 Å). Of course to derive the radial
velocity, the relativistic formula must be
employed however, in Cosmology the simple
connection between z and velocity breaks down,
because the expansion of the Universe is an
expansion of the metric itself, and it does not
reflect the motion of the source in a fixed
coordinate frame. Nor it is possible to reason in
terms of three spatial coordinates and one time
coordinate. For these reasons, the radial
velocities are better called indicative
velocities.
28
Arrossamento gravitazionale
General Relativity predicts another red-shift for
the light emitted by a source in a gravitational
field, e.g. by the surface of a star. At great
distance from a spherical source of mass M and
radius R, this gravitational red-shift is
expressed by
where G the gravitational constant, and rSch is
the Schwarzschilds radius. On the Sun, the
effect amounts to 0.64 km/s. On the white dwarf
Sirius B, having mass approximately equal to the
Suns, but radius of only 80 of that of the
Earth, the effect is correspondingly much larger.
The measurement of the gravitational red-shifts
from white dwarfs constitutes therefore another
test of the correctness of General Relativity.
29
Somma dei vari effetti
If a given object displays the kinematic,
cosmological and gravitational redshifts, the
combination of the different effects is given by
where the last approximate equality holds true
only for small redshifts.
30
Come si misura Vr - 1
Lets come back now to the practical task of
measuring radial velocities. First of all, the
observations must be corrected for the annual and
diurnal motions of the observer. The necessary
formulae are easily determined if the wanted
precision is around say 100 m/s. The heliocentric
velocity of the Earth varies between 29.3 km/s at
aphelion and 30.3 km/s at perihelion its
projection toward a direction of ecliptic
coordinates (?, ?) is
where VK 29.79 km/s, ?? is the longitude of
the Sun in that particular date, ?? is the
longitude of the perigee (approximately 18h48m),
and e the eccentricity of Earths orbit
(approximately 1/60). Notice that this term in
eccentricity, amounting to about 0.50 km/s, is
almost constant during the year.
31
Come si misura Vr - 2
Another way of expressing this correction is by
making use of equatorial coordinates of the line
of sight and of the Cartesian components of the
velocity, given in AU/day for each day by the
Astronomical Almanac
(km/s)
Regarding the diurnal rotation, the velocity of
the observer at the equator is approximately
0.465 km/s therefore, for a generic geocentric
latitude ? the projection of this velocity on
the line of sight to a star of declination ? and
Hour Angle HA will be
Should one need a precision better than 10 m/s
(very rarely achievable), then more accurate
formulae will be needed at this level, one
should also refer the velocities to the
barycentric, not to the heliocentric, observer
(the relative velocity with the heliocentric
observer being of approximately 12 m/s).
32
I moti propri - 1
Lets consider first the proper motion of the
nearer stars. Be S the heliocentric observer, and
X a generic star at a distance d with
heliocentric velocity V, at a certain date (see
Figure). Due to the enormous distances, apart
very few exceptions such as Barnards star, for
many decades or centuries the velocities can be
considered as rectilinear and uniform. Expressing
their modules in km/s, and indicating with n the
number of seconds in one year, after one year the
star will be seen in X, having traveled a course
of V?n km along a rectilinear path forming an
unknown angle ? with axis SX. On the plane
tangent to the celestial sphere, the star will
appear to have moved by the small angle
rad
33
I moti propri - 2
The component of V perpendicular to the line of
sight,Vt , is said transverse, or tangential,
velocity. The corresponding apparent angular
velocity ? is said proper motion of the star X,
and is commonly measured in arcsec/year, or
arcsec/century. By using the parallax ? in arcsec
instead of the distance d in km, and taking into
account the appropriate conversion factors
(namely 1 km/s 0.21095 AU/y, 1 AU/y 4.74045
km/s), we derive
km/s
arcsec/year

Barnards star has the highest proper motion,
10/y very few stars have ? larger than 2/year
obviously, the nearer stars have in general
larger proper motions, but the viceversa is not
true, many nearby stars have small motions
because of the orientation of their velocity
vectors.
34
I moti propri - 3
For the nearer stars, the annual ellipse of
parallax is trailed by the proper motion, as
evidenced in the Figure
The proper motion of ? 0.03 /year trails the
annual parallax ellipse of a nearby star having ?
0.08 (from Hipparcos the segments give the
instantaneous positions and associated errors,
the continuous line is the best fitting path) .
35
Moti propri più velocità radiali
If also the radial velocity of star X can be
measured from the Doppler effect
km/s
(the non-relativistic formula being certainly
valid), then the full velocity vector of the star
can be reconstructed. But for the majority of
cases, tangential angular velocities in arcsec
per year, and radial velocities in km/s, but not
the parallaxes, are available .
36
Moti propri come vettori - 1
It is convenient to consider the proper motion as
a vector ?(?, q) on the plane tangent to the
celestial sphere, with modulus ? expressed in
angular units (arcsec/year) along the great
circle XX, and direction expressed by a position
angle q measured from the North through the East
(0? q lt360). Alternatively, the two equatorial
components (??, ??) can be given attention must
be paid to the units of ??, because the angular
distance between two successive positions of the
star must be measured along the great circle
passing through XX, but quite often ?? is
derived by the difference between two successive
Right Ascensions.
37
Moti propri come vettori - 2
In the same manner, the tangential velocity
components are derived as
km/s
namely, the position angle of the proper motions
is the same as that of the tangential velocity,
and is independent of the parallax of the star.
Again with reference to Fig. 9. 3, from the
spherical triangle XNCPX we notice that
namely that the quantity is conserved during the
movement of the star, implying that
38
Effetto dei moti propri sulle coordinate
equatoriali
In order to find the effect of the proper motion
on the celestial coordinates of a given star, a
first order approximation will be sufficient for
short time intervals its mean coordinates T
years after the epoch t0 will be
a formula which includes the luni-solar
precession (the sign of T can obviously be
reversed), and with due attention to the units.
However, to be rigorous we must note that (??,
??) do vary in time even if the velocity is
rectilinear and constant. This is so for two
distinct reasons 1) the reference system rotates
because of precession 2) the changing
perspective alters the apparent length of equal
arcs, as shown (with great exaggeration) in the
Figure.
Perspective acceleration of a star in uniform
motion. The proper motion changes with time even
if vector V stays constant. The perspective
effect is obviously present also in the radial
velocity.
39
Derivate del moto proprio
Consider the successive terms in ?(t), ?(t)
The time derivatives of (??, ??) are composed of
two terms a) a term due to precession,
independent from radial velocity and distance,
and clearly the dominating one b) a term due to
the variation of the projection of the velocity
on the line of sight, and that can be written
down explicitly only if parallax and radial
velocity are both known. The components due to
precession clearly modify only the direction of
?, but not its modulus.
40
Variazione della distanza
Regarding the second, intrinsic, term, lets take
the time derivatives of eq., with the caution of
remembering that all derivatives must be in
circular units, for instance
Then
La seconda relazione può essere proiettata in AR
e DEC, purché si conoscano la velocità radiale e
la parallasse. Si noterà che se potessimo
misurare la variazione di parallasse avremmo una
misura della velocità radiale indipendente da
osservazioni spettroscopiche!
41
Relazione tra moti propri e costanti di
precessione
The previous discussion shows that the
uncertainties in the precessional constants will
enter into the uncertainty of the proper motion.
Such uncertainty is not so important for the
single star, but instead for the system of proper
motions. For instance, the FK5 could affected by
a spurious rotation at the level of about
0.15/century. This seemingly small systematic
error enters into the knowledge of the overall
field of motions and forces of the Milky Way.
Looking at the problem from the other side, a
reasonable model of the distribution of proper
motions can lead to the determination of the
precessional constants. Any effort must therefore
be made to obtain a system of proper motions as
precise as possible, and free from systematic
effect. The satellite Hipparcos could not produce
a major improvement, because its operational life
was too short. An alternative way is to derive
proper motions in respect to a non-rotating
background of fixed objects, such as the distant
quasars the reference system ICRF is by
definition devoid of rotation, so that many
efforts are presently made to refer the proper
motions to it.
42
Apice dei moti stellari - 1
Consider in Fig. 9. 5 the heliocentric equatorial
reference system, and be
the velocity components of star X.
On the celestial sphere, the direction of V
corresponds to point W (apex of the stellar
motion), and the great circle XW to the plane
passing by SX and containing the vector V.
43
Apice dei moti stellari - 2
These Cartesian components can be expressed in
terms of (Vt?, Vt?, Vr) by a suitable rotation
of coordinates. The Table gives the rotation
matrix that transforms (Vt?, Vt?, Vr) into
and viceversa.
for instance
44
Apice dei moti stellari - 3
Recalling the expression f the tangential
velocity Vt, we derive the Cartesian equatorial
components by the observable quantities
From the knowledge of the entire velocity vector
V, the direction of the motion of the star in the
heliocentric reference system S(x, y, z) can be
determined.
45
Coordinate dellapice W
The point W is called apex of the stellar motion,
and has equatorial coordinates given by
where (?, ?) are the initial coordinates of the
star. The angular distance between X and W is
given by
In general however, the parallax is unknown so
that the observations do not provide the length
of the arc ?. Nevertheless, the statistical study
of the stellar motions in certain areas of the
sky have evidenced the existence of stars having
motions converging towards the same apex
therefore these stars seem to belong to a group
having a common motion. We could therefore speak
of a co-moving group, or even of a stellar
current.
46
Lammasso aperto nelle Iadi
Lets consider in particular the very nearby open
cluster of the Hyades, whose distance is around
45 pc, and whose extension in space is
approximately 10 pc.
The convergence of the proper motions of the
stars to a common apex was ascertained long ago
(see Fig. 9. 6), but the data of the Hipparcos
satellite have provided a much improved knowledge
(see Perryman et al., 1998). This is one of the
few cases where the tri-dimensional structure can
be obtained directly from the observations, and
therefore the Hyades play a central role in the
calibration of several relationships, e.g. the
Hertzsprung - Russel (H-R) diagram.
47
Il moto peculiare del Sole
Siccome abbiamo riferito le velocità radiali e i
moti propri allosservatore solare, dobbiamo
attenderci che se questi ha un moto peculiare
rispetto allinsieme delle stelle vicine, tale
moto si trovi riflesso in qualche misura nelle
osservazioni. Esaminiamo dapprima i moti
propri. Already W. Herschel in 1783 had laid down
the foundations of the method, utilizing only the
position angles of 12 stars. Indeed, q is
independent from the parallax, and it coincides
with the position angle of the tangential
velocity. The method of Herschel can be easily
visualized draw on the celestial sphere the
great circles defined by the proper motions, and
consider the semi-circle oriented as the motion
itself. All these semi-circles will intersect,
within the errors, in a point (more
realistically, in a small area) which is the
antapex of the solar motion the modulus of the
velocity of the Sun will remain undetermined by
this method.
48
Un caso ideale per stelle vicine
Lets now examine the case of an assembly of N
nearby stars, for which the four quantities (?,?,
q, Vr), and therefore the heliocentric velocity
vector
(i 1,,N), are known. Lets calculate the
average value, change its sign and define this
quantity as the peculiar motion of the Sun in
respect to the given ensemble
The modulus s? and apex (??, ??) will be derived
in the usual manner
49
Il Local Standard of Rest (LSR)
The early observations provided a velocity of
approximately 20 km/s, in direction W?(??, ??) ?
(18h, 30?), not far from Vega. We stress that
this value depends on the particular set of stars
used to define it. Ideally, if we could take into
account all nearby stars, we would obtain a Local
Standard of Rest (LSR), a velocity reference
system of great interest for the study of the
velocity field of objects in the Milky Way. It is
therefore useful to transform the coordinate
system from the equatorial to the galactic one
(see Chapter 3) lets denote with (ui, vi, wi)
the three velocity components derived by
appropriate rotation of
with axis u directed toward the Galactic Center,
axis v at 90? in the galactic plane (toward the
constellation of Cygnus), and axis w toward the
galactic pole. Notice that this LSR has a purely
kinematics significance, the masses of the stars
not having been taken into account, and does not
posses a precise origin in space. We can assume
that the Sun (or better, the barycenter of the
Solar System) is passing by the LSR origin at the
present time.
50
Esercizi
1 - Discutere lequazione
dove ?? è la direzione tra la visuale e il
vettore velocità per diversi valori di ? tra 0
(velocità in allontanamento) e 180 (velocità in
avvicinamento).
2 - Reconstruct the velocity vector of the bright
star Capella, which has ? 0.075, ?
0.439/year, Vr 30.2 km/s. The vector V is
directed to ? 42.5 deg, with modulus of 41.0
km/s. 3 Trovare sul sito del satellite
astrometrico Hipparcos le 20 stelle di maggior
parallasse e le 10 stelle di maggior moto
proprio. I due insiemi coincidono?