Download Obliquity and precession of the equinoxes The angle ε between the

Astronomical background II
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Obliquity and precession of the equinoxes
The angle ε between the ecliptic and the celestial
equator is called the angle of obliquity. It is the
angle between the fundamental planes of these two
coordinate systems, that is, between the Earth’s
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orbital plane and its equatorial plane. Therefore, it
is the same as the angle between the ecliptic polar
axis (the direction perpendicular to Earth’s orbit)
and the Earth’s axis (which is perpendicular to its
equator); it is a measure of Earth’s axial tilt. We
know this angle to be about ε = 23°26 ′ . It is this
angle that controls the climatic variation of the
seasons on Earth and helps to define the Tropics of
Cancer and Capricorn.
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But in the same way that the spin axis of a top
wobbles about as it spins across a flat surface, the
Earth’s axis also wobbles relative to its orbital
plane, so that the polar point P describes a small
circle around the north ecliptic pole (in Draco).
This motion is extremely slow, making its way once
around this circle every 25,800 years, but it has the
effect of sliding the north celestial pole P away
from Polaris over time. This means that the north
celestial pole was much closer to the dim star
Thuban ( α Draconis) in 3000BCE, and will drift
towards another dim star, γ Cephei, in 4000CE.
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It also has the effect of sliding the celestial equator
around the ecliptic, changing the position of the
First Point of Aries in the celestial sphere, by about
50 seconds of arc per year or by 30° every 2150
years. This motion of the Earth is called the
precession of the equinoxes. During the first
millenium BCE, during the height of Babylonian
astronomy, the First Point of Aries actually was
found within the constellation Aries, but in time it
drifted into Pisces. Currently we find ourselves at
“the dawning of the Age of Aquarius”, where the
First Point of Aries will travel for the next 2150
years or so.
In addition to precession, the circular path of the
north celestial pole P undergoes a small wiggling
motion called nutation [L. nutare = to nod], which
causes P to oscillate back and forth every 18.6
years (6798 days) as it precesses. This motion is
due to tidal forces among the planets and the Sun.
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Solar and sidereal days
Local noon occurs when the Sun makes its upper
transit, crossing the meridian. For observers in the
Northern Hemisphere, this is when the Sun is
furthest south in the sky (and when shadows point
north and are the shortest). The time between two
consecutive local noons is the solar day. Of course,
the solar day marks one rotation of the Earth on its
axis. But as we have already noted, the Sun also
moves against the celestial sphere by a small
distance (about 1° of right ascension) each day; that
is, the Earth moves a small distance along its orbit
about the Sun during a day. As a result, when the
Earth makes one 360° revolution about its axis, the
Sun is no longer in the same position it held on the
celestial sphere at midday on the day before: it is 1°
further to the east. It must rotate a further 1° to
complete the solar day. Consequently, the period of
time it takes for the Earth to make one full
revolution of its axis is less than one solar day.
However, it is the same as the time between
successive transits of any star in the celestial
sphere. We call this the sidereal day [L. sidus =
constellation].
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How long is the difference between the solar and
sidereal day? We know that in the span of a year,
there are 365.25 solar days. But in that same time,
there must be 366.25 sidereal days, since in the
time it takes the Earth to go completely around the
Sun, it rotates against the celestial sphere one
more time, the accumulation of all the tiny deficits
in rotation after each solar day throughout the
year. Thus,
366.25 sidereal days = 365.25 solar days
365.25
1 sidereal day =
⋅1 solar day
366.25
365.25
=
⋅ 24 h 0 m 0 s
366.25
= 23 h 56 m 4 s
meaning that the sidereal day is 0 h 3 m 56 s shorter
than the solar day. This explains why stars that
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are seen to rise on a particular night at a particular
time will be seen to rise the next night 4 minutes
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earlier. The cumulative effect over time of this
phenomenon is to change the set of constellations
that are visible at the same time every night. Over
the course of the year, each constellation graces the
night sky for a time, rising earlier and earlier, then
vanishing in the sunlight to return again the next
year.
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The equation of time
We generally believe that the solar day equals a
constant 24 hours. But, surprisingly, the solar day
varies in length across the year, by as much as 50
seconds at its extremes! That the length of the day
was not constant was recognized even by ancient
astronomers, who observed it by recording the
behavior of shadow clocks over the course of a full
year. But these effects were not large enough to
concern them, and their tools were not sensitive
enough to measure the effects accurately until the
development of reliable clocks that marked time in
constant units in the 17th c. The use of such clocks
made it possible to fix the length of the day at 24
equal hours; however, this period of time is actually
the mean solar day, which averages out the
annual variations in the apparent solar day.
The length of the apparent solar day varies for two
main reasons, the effects of Earth’s elliptical orbit
and the obliquity of the ecliptic. Let’s explain this
more fully.
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(1) Kepler’s Laws state that a planet moves more
quickly when it is closer to the Sun in its elliptical
orbit (Earth reaches perihelion [Gr. peri + helios =
around + Sun], its point of closest approach, in
early January, and aphelion [Gr.apo + helios =
away from + Sun] in early July). Consequently,
winter solar days should be slightly shorter than
summer solar days. The Sun transits precisely at
noon on local clocks on those days, but for many
days after perihelion, the cumulative effect of
consecutive short solar days causes the transit to
occur earlier and earlier on the clock since the clock
continues to measure out a constant 24 hour day.
By the end of March, local noon arrives at about
11:53am. This effect reverses as the Sun
approaches aphelion, but then after aphelion, the
cumulative effect of consecutive long solar days
causes the transit to occur later and later by the
clock so that by the end of September, local noon
occurs at 12:07pm. Thereafter, the trend reverses
yet again. This process repeats every year.
(2) Even if the Earth’s orbit were perfectly
circular, the Sun would still move along the ecliptic
a small equal amount every day. But as the
ecliptic is tilted with respect to the celestial
equator, the Sun moves a shorter distance of right
ascension on the days of the equinoxes (since the
ecliptic is most angled against the equator then);
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the more slowly moving Sun makes for longer than
average days then (by as much as 20 sec per day).
The Sun moves a longer distance of right ascension
on the days of the solstices (since the ecliptic is
parallel to the equator then); the more quickly
moving Sun makes for shorter than average days
(again, by as much as 20 sec per day). Therefore,
since the Sun would transit at 12:00 noon on the
vernal equinox, the progressively longer solar days
that follow would cause the transit to occur later
and later by the clock so that by the cross-quarter
day at the beginning of May, the Sun would transit
at about 12:10pm. As the days shorten moving
towards the summer solstice, the midday transit
begins to occur earlier and earlier, so than by the
beginning of August, it occurs at about 11:50am.
Then the effect reverses again through the
autumnal equinox (when local and solar noon
agree) to reach a second turning point around the
beginning of November, when solar noon is at
12:10pm. This oscillation continues through the
winter, when at the fourth cross-quarter day in
early February, solar noon occurs its earliest, at
11:50am.
These two phenomena, the effect of the Earth’s
elliptical orbit and the obliquity of the ecliptic,
actually work in concert to cause a regular and
annual variability in the length of the solar day.
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The difference between apparent local noon (the
time of the Sun’s transit) and mean local noon
(when a clock set to local time would read 12:00) is
called the equation of time. Controling for this
variation is important if one wants a sundial, for
instance, to synchronize time with a watch.
Finally, the sidereal day, which is 23 h 56 m 4 s long,
is also undergoing very slight long-term changes,
called secular effects [L. saeculum = age or era].
Tidal gravitational forces cause the Earth’s
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rotation to slow down slightly, lengthening the
sidereal day by about 2 milliseconds (0.002 sec)
every century, a very small change, to be sure.
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Definitions of the year
The ancient Chinese marked off the years by
determining the solstices from shadow clocks: the
summer solstice occurred when midday shadows
were shortest, the winter solstice when they were
longest. The Incas, living in the tropics, used the
absence of shadows at noon shining through a long
tube to mark the annual day when the Sun was
directly overhead at midday. And the Egyptians
used the heliacal rising of Sirius to time the
beginning of their year. In all these cases, the year
was found empirically. As astronomical knowledge
improved, it became possible to identify more
accurately not only how long the year was, but
what phenomena controlled its passing.
As the year was typically tied to seasonal events,
the earliest understanding led to the definition of a
year as the time between consecutive summer (or
winter) solstices (or between consecutive vernal or
autumnal equinoxes; all four definitions are
equivalent). It was known from prehistoric times
that this is often 365 days, and sometimes 366.
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Modern astronomers now define the tropical year
to be the time it takes the Sun to return to the First
Point of Aries along the ecliptic. (This is
essentially the same time it takes for the Sun to
return each year to the position of the Tropic of
Cancer at the summer solstice.) To 10 decimal
places of accuracy,
1 tropical year = 365.2421896698 (mean solar) days
However, there is one important flaw in using this
event to define the year. Nutation and precession
of the equinoxes causes the First Point of Aries to
shift over time along the celestial equator. This
causes the tropical year to deviate by about 10
minutes’ time (or 0.014 days) from the value above
from year to year, so the above must be considered
a mean value. Other secular effects involving the
gravitational interaction of the Earth with other
bodies in the solar system are working to shorten
the length of the year very slightly over time: the
above mean value is now 12 seconds shorter than it
was 2150 years ago when the Greek astronomer
Hipparchus made the first serious attempt to
measure it. (See Table 2.3, p. 33, for a table of
historical measurements of the tropical year.)
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The lengths of the seasons
By Kepler’s Laws, we know that the Earth’s orbit is
elliptical, reaching perihelion in early January
when the Earth travels fastest around the Sun, and
aphelion in early July when the Earth travels most
slowly. It follows that the four seasons, which have
the solstices and equinoxes as their division points,
are not of equal duration.
In particular, winter (in the Northern Hemisphere)
is the shortest season and summer is the longest:
Season
Begins at
Approx. date Length (days)
Spring
vernal equinox
Mar 21
92.8
Summer
summer solstice
Jun 21
93.6
Fall
autumnal equinox
Sep 22
89.9
Winter
winter solstice
Dec 21
89.0
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The month and the Moon’s motions
The Moon orbits the Earth. It is new when it is in
conjunction, that is, when it has the same ecliptic
longitude as the Sun. In this geometry, the light of
the Sun falls on the side of the Moon facing away
from the observer. This also means that the Moon
must be near the Sun in the sky, so the Moon at
conjunction is not easy to see, except just before
sunrise or after sunset (or when it passes directly in
front of the Sun during a solar eclipse!).
Likewise, the Moon is full when it is in
opposition, that is, when its ecliptic longitude
differs by 180° with that of the Sun. Here, it is
roughly in line with the Earth and Sun, but the
light of the Sun strikes the side of the Moon facing
the observer. Here the Moon is easy to see, as it
stays in the sky all night. When the Moon passes
directly in the path of the Sun and the Earth, a
lunar eclipse takes place and the face of the Moon
is temporarily blocked by the Earth itself.
The time between successive conjunctions is one
lunation, also called a synodic month [Gr.
synodos = meeting]. Lunations vary in length by as
much as 3.5 hours from the mean synodic month of
29.53059 days. Tiny secular effects are increasing
the synodic month by about .02 sec per century.
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In the same way that the sidereal day varies from
the solar day, a sidereal month, the time it takes
for the Moon to reoccupy the same position on the
celestial sphere, differs from the synodic month. It
is a bit shorter, 27.32166 days, and for a similar
reason that tells why a sidereal day is shorter than
a solar day.
The orbit of the Moon is in a plane that is tilted
with respect to the plane of the Earth’s orbit by
5°8´. This explains why there aren’t a pair of
eclipses every month: the Moon only crosses the
ecliptic twice each month, at a pair of points called
the lunar nodes. As it crosses the ascending
node, it moves above the ecliptic, and when it
crosses the descending node, it goes below the
ecliptic. Only when these crossings occur at
conjuction is there is a solar eclipse, and when they
occurs at opposition, there is a lunar eclipse. But
the nodes themselves rotate through the ecliptic
over time, with a period of 18.6 years (called
precession of the nodes), so eclipses occur only
infrequently.
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The time between successive passages of the Moon
through its ascending node is called the draconic
month; this term derives from the astrological
practice of labeling the ecliptic longitude of the
ascending lunar node (which varies over time) with
the symbol , called the dragon’s head. (The
ecliptic longitude of the descending node was
denoted , the dragon’s tail.) The draconic month
is 27.21222 days, slightly shorter than the sidereal
month, since the nodes rotate to meet the orbiting
Moon.
The 5° tilt also allows the Moon to rise a bit higher
than or fall a bit lower than the Sun along the
ecliptic. Therefore, its declination reaches a
maximum of δ = 28°34′ = 23°26 ′ + 5°8 ′ and is highest
in the sky when the Moon is halfway between
ascending and descending nodes at a summer
solstice. This event is called the major lunar
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standstill (analogous to a solstice event for the
Sun). Two weeks later, the Moon will be in the
opposite position in its orbit, so it will reach a
(near) minimum declination of δ = −28°34 ′ .
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Ancient Babylonian astronomers discovered that,
although the synodic month and draconic month
are unequal, their cycles synchronized every 18
years or so. Since the synodic month determined
the cycle of new and full moons and the draconic
month determined the cycle of passage through the
nodes, the combination of these cycles controlled
the pattern of eclipse events. Specifically, they
knew that the synodic month was 29;31,50 days
long (sexagesimal notation) and the draconic month
was 27;12,42,30 days long, and noticed that
223 synodic months = 223 × ( 29; 31, 50 days)
= 6585;18, 50 days
was roughly equal to
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242 draconic months = 242 × ( 27;12, 44 days)
= 6585; 21, 28 days
(The difference of 0;2,38 days equals about 1 hour.)
This period of 6585 13 days, or 18 years plus 11 13
days, came to be known the saros.
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Any solar or lunar eclipse event would then be
followed 6585 13 days later by another of the same
type (although not in the same geographical
location: the extra 13 day means that the event
takes
place 13 of the way around the Earth from
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where it occurred in the previous cycle).
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For instance, the most recent solar eclipse in
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Cincinnati
occurred on May 10, 1994. Another
solar eclipse will follow on May 20, 2012 (18 years
11 days later) at roughly the same latitude but 120°
of longitude further west, in the North Pacific
(visible in the Aleutian Islands).
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Finally, we recognize that the orbit of the Moon is
not circular, but elliptical, like that of each of the
planets about the Sun. So it comes closest to the
Earth at the point of perigee [Gr. peri + ge =
around + Earth], and is furthest away at another,
the point of apogee [Gr. apo + ge = off of + Earth].
These two points are together called the Moon’s
apsides (sing. apsis) [Gr. apsis = the point on a
wheel where the two ends are fastened together].
The line of apsides connects the points of perigee
and apogee and, because of solar gravitational
forces, it precesses about the Moon, making a
complete rotation once every 8.85 years. At
perigee, the Moon is a bit larger in appearance
( 0°34′ wide) to Earth observers than at apogee
( 0°29′ wide).
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The Moon’s anomaly [Gr. anomalia = irregularity]
is its angular measure east of perigee along its
orbit, so the anomaly at perigee is 0° and at apogee
is 180°. The time between successive perigee
positions is therefore called the anomalistic
month; it is 27.55455 days, a bit longer than the
synodic month since the line of apsides rotates
ahead of the Moon in its orbit.
The saros is also conveniently equal to about 239
anomalistic months, so the size of the Moon at
successive eclipses in a saros cycle is roughly equal.