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Changes of Celestial Coordinates
Several effects lead to the notion that measured coordinates slightly
deviate from those given, e.g., in catalogs, and that these measurements
change with time, location and observing conditions. The only "real"
effect, i.e. physical change of relative position, is the proper motion
of solar system objects and stars, which causes the objects to come into
another direction (and distance) with respect to Earth.
Effects on Earth's motion leads to coordinate effects which don't change
any direction from the starlight coming, but of the measurement (or the
coordinate system) only: precession and nutation.
The motion of Earth with a periodically varying direction of velocity is
also responsible for the aberration of light, an apparent deviation of
stars from their position with annual periodicity.
Finally, nearby objects in the solar system show positional and daily
parallax effects, and similarly, parallax can be measured for nearby stars.
Besides proper motion, this is the only effect discussed here showing up,
e.g., in photographs.
At last, the Earth's atmosphere makes stars to appear in another position
than they are, this is called refraction.
Historically, precession was already discovered around 130 BC by ancient
Greek astronomer Hipparchus when he compared his observations with those of
predecessors from about 200 years earlier. Edmond Halley discovered the
proper motion of stars in 1718 when comparing his observations with Ptolemy's
catalog (based on Hipparchus' observations), James Bradley detected the
aberration of light in 1725-26 (published 1729) and the nutation of Earth
in 1747, while the parallax of fixed stars was not detected before 1838
when Friedrich Wilhelm Bessel discovered it for 61 Cygni.
Stars are not really fixed in space but move according to their space
velocity and the gravitational field in their environment. As a star changes
its absolute position in space, it will slowly change the direction in which
it appears to be from Earth (which also changes its position as the Solar
System moves through space). This will be visible as a continuously growing
displacement of the star from its original position.
Only the tangential component of the relative motion of a star shows up
in proper motion, which is measured in arc seconds per year or per century;
the radial component (which changes the distance) can be measured with
much higher accuracy in the Doppler shift of spectral lines visible in the
spectra of stars.
The star with the largest observed proper motion is 9.7 mag Barnard's Star in
with 10.27 "/y (arc seconds per year). According to F. Schmeidler, only about
500 stars are known to have proper motions of more than 1 "/y.
Precession of the Earth's polar axis is caused by the gravitational pull of
the Sun and the Moon on the equatorial bulge of the flattened rotating Earth.
It makes the polar axis precess around the pole of the ecliptic, with a
period of 25,725 years (the so-called Platonic year). The effect is
large enough for changing the equatorial coordinate system significantly
in comparatively short times (therefore, Hipparchus was able to discover it
around 130 B.C.). Sun and moon together give rise to the lunisolar
precession p0, while the other planets contribute the
significantly smaller planetary precession p1, which sum up
to the general precession p; numerical values for these
quantities are (from Schmeidler; t is the time in tropical years
p0 = 50.3878" + 0.000049" * t
p1 = - 0.1055" + 0.000189" * t
p = 50.2910" + 0.000222" * t
These values give the annual increase of ecliptical longitude for all stars.
The effect on equatorial coordinates is formally more complicated, and
approximately given by
p_RA = m + n * sin RA * tan Dec
p_Dec = n * cos RA
where the constants m and n are the precession components given by
m = + 46.124" + 0.000279" * t
= 3.0749 s + 0.0000186 s * t
n = + 20.043" - 0.000085" * t
= 1.3362s - 0.0000056 s * t
Precession will shift the North Celestial Pole even closer to Polaris
(Alpha Ursae Minoris) until 2115, and around 14,000 AD, Vega (Alpha Lyrae)
will be an extremely bright polar star.
About 5000 years ago, closest to the celestial pole about 2850 BC, Thuban
(Alpha Draconis) has been the pole star.
As Earth's axis precesses around the pole of the ecliptic, this motion
is superimposed by small periodic fluctuations called nutation.
Ths nutation is caused by the motion of Lunar orbital nodes, which is
retograde and has a period of 18.60 tropical years. Due to this effect,
the celestial poles follow small ellpises with a semimajor axis of 9.202",
which is called constant of nutation.
As Earth moves through space, starlight seems to come from a different
direction as if Earth were at rest; this effect is called the
(annual) aberration of light.
It causes stars to be displaced along an ellipse of
a = k = 20.496" semimajor axis (this is the aberration constant), with
semiminor axis depending on ecliptical latitude be: b = k * sin be.
Similarly, slower Earth's rotation causes the
diurnal aberration of light with aberration constant 0.31".
Daily and annual motion of an observer due to Earth's rotation and revolution
causes diurnal and annual parallax, a slight displacement of
objects which are not too far away compared to Earth's diameter or the
diameter of Earth's orbit, respectively.
When light passes from one medium to another medium of different density
(e.g., from the vacuum to Earth's atmosphere), the speed of light in the
medium is changed (light is slower in the denser medium), causing the
wavelengths of light to bend at different angles.
Refraction in Earth's atmosphere of light coming from a celestial object
causes the object to appear in a slightly shifted position than it
actually is; more acurately, objects near the horizon appear "lifted" to
slightly higher altitudes. This effect even lifts objects from up to
35 arc minutes below horizon to above the apparent horizon, so that they
can be seen, and it makes objects apparently rise a bit earlier and set
a bit later.
- Diurnal parallax:
- Daily displacement is maximal at the equator, so that the parallax
pi is maximal there. Its amount
pi0 is given by sin pi0 = R/d for a body at distance
d, with equatorial Earth radius R. For other
geographical latitudes B, the right hand side of this relation is to be
multiplied by a factor cos B.
For the Moon: pi0 = 3422.44" = 57' 2.44" (almost 1 deg)
For the Sun: pi0 = 8.794"
- Annual parallax:
- In this case, Earth's orbit is the baseline for the parallax construction.
The effect is very small, as stars are far away, the nearest being Alpha
Centauri with an annual parallax of 0.772" (distance 4.3 light years).
While almost always unimportant for coordinate measurements, this effect
is of vital importance for the
determination of distances
in the universe.
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