{Note: If your browser does not distinguish
between "a,b" and "α, β" (the Greek
letters "alpha, beta")
then I am afraid you will not be able to make much sense of the
equations on this page.}
So far, this series of pages has considered The Earths axis is tilted to its orbital plane.
Around 2000 years ago, Exercise: The Vernal Equinox occurs
nowadays Pisces covers a section of the
ecliptic How many years would we have to
go back, Click here
for the answer. Precession is caused by the Sun and the
Moon. Ignoring nutation, This definition assumes the ecliptic itself is
unchanging. Combining luni-solar and planetary precessions gives
general precession. Because of precession, To point a telescope at an object Recall the formulae relating equatorial and ecliptic
coordinates:
Luni-solar precession affects the ecliptic longitude
λ. Consider luni-solar precession first, To find how the declination δ
changes with time t, To find how the Right Ascension α changes with time,
So if Δλ is the change in λ in a given
time interval Δt,
This is the effect of luni-solar precession. The combination is general precession:
To make this easier to calculate in practice, These quantities m and n are almost constant; We can now write:
which means that, Exercise: The coordinates of the Galactic
North Pole are given officially as What should they be, (For this calculation, take the
values of m and n for the year 1975: Click here
for the answer. Alternatively, the Astronomical Almanac lists
Besselian Day Numbers throughout the year. Previous section:
Aberration
how we
assign coordinates to any point in the sky,
and the various
physical effects that may alter its apparent position.
But there
is a more profound problem with the way we determine
coordinates,
relative to the celestial equator and the
ecliptic,
since these are not permanently fixed.
The gravitational pull of the Sun and the Moon on the Earths
equatorial bulge
tend to pull it back towards the plane of the
ecliptic.
Since the Earth is spinning, its axis precesses.
The North Celestial Pole traces out a precessional circle
around the pole of the ecliptic,
and this means that the
equinoxes precess backwards around the ecliptic,
at
the rate of 50.35 arc-seconds per year
(around 26,000 years for a
complete cycle).
the Sun was in the
constellation of Aries at the spring equinox,
in Cancer at the
summer solstice,
in Libra at the autumn equinox,
and in
Capricorn at the winter solstice.
Precession means that all of
these have changed,
but we still use the old names
(e.g. the
First Point of Aries for the vernal equinox),
and the symbols for
the vernal and autumnal equinoxes
are the astrological symbols
for Aries and Libra.
when the Sun is in the constellation of Pisces.
from longitude 352° to longitude 28°;
at
longitude 28° the ecliptic passes into Aries.
to find the Sun at the First Point of Aries
at the vernal equinox?
However, the Moon does not orbit exactly in the ecliptic
plane,
but at an inclination of about 5° to it.
The
Moons orbit precesses rapidly,
with the nodes taking 18.6
years to complete one circuit.
The lunar contribution to
luni-solar precession
adds a short-period, small-amplitude
wobble
to the precessional movement of the North Celestial
Pole,;
this wobble is called nutation.
luni-solar precession simply
adds 50.35 arc-seconds per year
to the ecliptic longitude of
every star,
leaving the ecliptic latitude unchanged.
In fact, the gravitational pull of the other planets
perturbs the Earths orbit
and so it gradually changes the
plane of the ecliptic.
If the equator were kept fixed,
the
movement of the ecliptic would shift the equinoxes forward along the
equator
by about 0.13 arc-seconds per year.
This is planetary
precession,
which decreases the Right Ascension of
every star by 0.13 arc-seconds per year,
leaving the declination
unchanged.
(Lunar nutation and planetary
precession also produce slight changes in the obliquity of the
ecliptic)
our framework of Right
Ascension and declination is constantly changing.
Consequently,
it is necessary to state the equator and equinox
of the
coordinate system to which any position is referred.
Certain
dates (e.g. 1950.0, 2000.0) are taken as standard epochs,
and used for star catalogues etc.
on a date other
than its catalogue epoch,
it is necessary to correct for
precession.
sin(δ) = sin(β)
cos(ε) + cos(β) sin(ε) sin(λ)
sin(β) = sin(δ)
cos(ε) - cos(δ) sin(ε) sin(α)
cos(λ) cos(β) = cos(α)
cos(δ)
The resulting corrections to
Right Ascension and declination
can be worked out by spherical
trigonometry.
But here we use a different technique.
recalling
that it causes λ to increase at a
known, steady rate dλ/dt,
while
β and ε
remain constant.
take the first equation and differentiate
it:
cos(δ) dδ/dt = cos(β)
sin(ε) cos(λ) dλ/dt
To eliminate β and λ from this equation,
use the third equation:
cos(δ) dδ/dt = cos(α)
sin(ε) cos(δ) dλ/dt
i.e.
dδ/dt
= cos(α) sin(ε) dλ/dt
take the second equation and differentiate it:
0 = cos(ε) cos(δ)
dδ/dt + sin(ε) sin(δ) dδ/dt sin(α) -
sin(ε) cos(δ) cos(α) dα/dt
i.e. sin(ε) cos(δ)
cos(α) dα/dt = dδ/dt [ cos(ε) cos(δ) +
sin(ε) sin(δ) sin(α) ]
 
; =
cos(α) sin(ε) dλ/dt [cos(ε) cos(δ) +
sin(ε) sin(δ) sin(α) ]
Cancelling out sin(ε) and cos(α) from both sides gives:
cos(δ) dα/dt = dλ/dt
[ cos(ε) cos(δ) + sin(ε) sin(δ) sin(α) ]
Dividing through by cos(δ) gives:
dα/dt = [ cos(ε) +
sin(ε) sin(α) tan(δ) ] dλ/dt
the corresponding changes in α and δ are
Δα = Δλ [
cos(ε) + sin(ε) sin(α) tan(δ) ]
Δδ = Δλ
cos(α) sin(ε)
We
also have to add in the planetary precession,
which decreases the
RA by a quantity a, during the same time interval.
Δα = δλ [
cos(ε) + sin(ε) sin(α) tan(δ) ] - a
Δδ = Δλ
cos(α) sin(ε)
we
introduce two new variables, m and n:
m = Δλ cos(ε)
- a
n = Δλ
sin(ε)
they
are given each year in the Astronomical Almanac.
The
values for 2000 are approximately:
m
= 3.075 seconds of time per year
n
= 1.336 seconds of time per year
=
20.043 arc-seconds per year
Δα
= m + n sin(α) tan(δ)
Δδ = n
cos(α)
if you know the equatorial
coordinates of an object at one date,
you can calculate what they
should be at another date,
as long as the interval is not too
great (20 years or so).
If the object is a star whose proper
motion is known,
then that should be corrected for as well.
α
= 12h49m00s, δ = +27°24'00",
relative to the equator and equinox of 1950.0.
relative
to the equator and equinox of 2000.0?
m = 3.074s per year;
n =
1.337s per year = 20.049" per year.)
Take a stars
equatorial coordinates from a catalogue,
and compute various
constants from these,
as instructed in the Astronomical
Almanac.
Combine these with the Day Numbers for a given date,
to produce the apparent position of the star,
corrected for
precession, nutation and aberration.
Next section: Calendars
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