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FOUR
CARDINAL STARS
Markers
of the Great Cycle
-
Part One -
Haluk Akcam
Jan.
01, 2004 |
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Great
Cycle
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A
full precession of the equinoxes is known as the Great Cycle, of about
25,800 years, and regarded mostly in respect to cycle analyses.
Precession is the movement of the First Point of Aries backwards
along the equator at about 50.3 arcseconds per year, due to the conical
motion of the Earth's rotation axis caused mainly by the gravitational
pull of the Sun and Moon on the Earth's equatorial bulge. Nutation
is the wobble of the Earth's axis while precessing.
It
is not a fixed duration, and varies according to the period, which is
under consideration. For the present epoch (J2000.0), value of the
general precession rate is about 50.288 arcseconds per year, causing one
degree retrograde motion of the equinoxes in round 71.6 years, and
defining the instantaneous length of the Great Cycle as 25,771 years, accordingly. In tradition, length of the
Great Cycle is 25,920 years, with 50.00 arcseconds per year, or one
degree in 72 years.
General
precession rate is a practical but not a sufficient parameter to obtain
numerical values for long-term intervals. This formula may serve as a good approximation to estimate the
accumulated precession:
p
= 5028.792262 T1 + 1.1124406 T2 + 7.699×10-5
T3 - 2.3479×10-5 T4 - 1.78×10-8
T5 + 1.8×10-10 T6 + 1×10-12 T7
Tn
= [ ( t2 - 2000 ) / 100 ]n
- [ ( t1 - 2000 ) / 100 ]n
Where
t1 and t2, initial and terminal epochs of the
interval, are counted as Julian years, to obtain
the general precession amount in seconds of arc. The
above polynomial is adapted by me after the expressions for precession proposed
by Bretagnon et al.[1], which seems to be the best available one at the
moment, for calculating the partial accumulated general precession in
longitude, for a given interval.
Yet, the validity of this formula must be restricted to the period 7635
BC - 12250 AD. Within these limits, the annual precession rate varies
between 48.895 and 51.686 arcseconds, which corresponds to 26505 and
25074 years. Therefore, I should remind the reader that since the astronomical
simulation programs are not using a better computation, what they claim
of accuracy does not have any validity beyond these maximal limits.
Determining
the Beginning of a Cycle
With
our present knowledge, it is neither possible to give the exact date of
the beginning or the end of a precession cycle nor the exact length of a
cycle, for remote dates. Yet, we may assign a certain event or chain of events to the
beginning of a cycle or a phase, provided that the event is bearing
an universal importance. Such attempts have been practiced by some
astrologers, by emphasizing the most important event related to their
culture, like the birth of a religious figure or the rise of an empire.
But, these attempts are mostly subjective in nature, lacking any
astronomical support, as the experience shows.
For
instance, due to the expand of astrology in western cultures, the birth of
Jesus Christ is regarded as a great omen and a mark of the so
called Piscean Age (one of the twelve fictitious ages of
the precession cycle, named after the twelfth sign of the astrological
zodiac), which is supposed to last 2160 years! But, the astronomical
background of this assumption is totally void, except the very suspicious
legendary triple conjunction of some visible planets or objects, which occurs
certainly more than once in two millenniums. On the other hand, for the Far
East, the birth of Buddha may seem more important, or the birth of Muhammad
for the Muslims. But, the time intervals between the appearance of these
religious figures do not support this hypothesis, even if more samples can be
introduced.
The
idea to divide one precession cycle into twelve equal sequences has its
origin in the fictitious division of the zodiac. But, since the zodiac is a
belt fitted to the apparent path of the sun, merely known as the ecliptic
circle, we simply cannot assign twelve arbitrary sectors without any
astronomical verification behind it. Yet, this division has some relevance to
the astronomical facts: The two equinoctial points
and the two solstices are marking a cross on the ecliptic plane, where each
sector can be divided further into three equal parts. But, this secondary
division then becomes arbitrary. For instance, anyone else may prefer to
divide these sectors into four, or six equal parts, as long as no astronomical
relevance is required. Therefore, if an analogy is sought, one precession
cycle may be divided into four sequences, but not any more!
On
the other hand, fourfold division is useful to understand the implications of
this great cycle, and therefore becomes essential in the examination, due to
the changing effect of the Sun, when the point of periapsis coincides one of
these four points. It is therefore called by the ancients "the four
cardinal points" and the related "four cardinal stars",
in remote antiquity. But, further arbitrary divisions are only good for
producing some speculations, and we do not see any sample of it in the inherited
old tradition, which is certainly not soothsaying but merely the
foundation of today's astronomy yet for the benefit of mankind.
The
coincidental relation between the point of periapsis and the four cardinal
points has been examined mostly to predict large scale climatic changes,
especially after the theory of Milankovitch. Our subject in this article is
not the climatic history of the earth, but the cultural one. For this purpose,
one of the remnants of the forgotten faith is selected, namely the four
cardinal stars of the ancient lore.
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Obliquity
of the Ecliptic
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Obliquity
of the ecliptic is defined by the angle between the rotational and orbital
planes of the Earth. Just like the precession rate, it is not a fixed
angle. It is
supposed to vary as follows: (t as Julian centuries since J2000.0, and the
product is in arcseconds)
e
= 84381.4088 - 46.836051 t - 1.667×10-4 t2 +
1.99911×10-3 t3
- 5.23×10-7 t4 - 2.48×10-8
t5 - 3×10-11 t6
Again,
this polynomial definition has limits in application: Between 7452 BC
(24°14') and 12608 AD (22°36'), it works sufficiently. But beyond these
limits, the value is becoming highly deviated. This one is the best
expression of the obliquity proposed by Bretagnon [1] among the recent
models.
A
detailed comparison of the numerical expressions for the precession and the
obliquity is discussed in a separate article.
Although highly precise for a few millenniums, they are
not much of a value for long-term calculations, as the following graphs show
the comparison. Each vertical gray time grid interval represents 1000 Julian
years. The sinus curve is
representing the possible oscillator adapted by applying the numerical
expressions of the polynomial.
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Obliquity of the
Ecliptic
Yellow
curve is according to the polynomial proposed by Bretagnon (2003).
Red
curve is the approximated sinusoidal. |
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General Precession
rate
Yellow
curve is according to the polynomial proposed by Bretagnon (2003).
Red
curve is the approximated sinusoidal. |
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A
simple comparison shows that the precession rate proportionally increases when the
obliquity decreases, or vice versa. Since the obliquity angle varies
between 22.0° and 24.6°, the instantaneous length of the great cycle
is estimated round 24500 - 26800 years. The fluctuation period of the
obliquity is round 41000 years. |
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Four
Cardinal Stars
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According
to the tradition, the Great Cycle is marked by four stars located at the
cardinal points of the heaven. The first and the famous of these four is
Regulus. Then comes the opposing pair Antares and
Aldebaran. The fourth is somewhat ill-defined, but supposed to be
Fomalhaut. In fact, they are not exactly forming a cross, but
throughout the history, they are known as the four royal or cardinal
stars.[2][3]
Regulus
(a Leonis) gets its name from the note of
Copernicus; "in corde quam Basiliscum siue regulum uocant",
where he mentioned BasiliskoV of the
ancients. Arabs named it as qalb al-asad (heart of the lion), but
western astronomers preferred the definition of Copernicus.
Aldebaran
(a Tauri) is directly named after the
Arabic definition ad-dabaran (follower), which originates
probably from the translation of Ptolemy's explanation, where he cites
the star next to Hyades (lamproV tvn ¢Uadwn).
Anyway, the meaning of the Arabic word is dubious.
Fomalhaut
(a Piscis Austrini) is again directly
named after the Arabic definition. It is the fam al-hut al-janubi
(mouth of the southern fish), an exact translation of the definition of
Ptolemy. The name appeared in the history of astronomical literature
with variants, but the origin is the same.
Antares
(a Scorpionis) is originated from the
description of the star as AntarhV (like
Mars, in the nature of Mars, but sometimes as rival of Mars). Ptolemy named it as the heart of the
scorpion, so did the Arabs (qalb al-aqrab). But the description
substituted the name and remained until today.
Here
is some astronomical data from reliable recent catalogues:
| Name |
HIP |
a
(arcdeg) |
d
(arcdeg) |
ma*
(mas/yr) |
md
(mas/yr) |
Vm |
| Regulus |
49669 |
152.09358075 |
+11.96719513 |
-248.7 |
+5.3 |
1.360 |
| Aldebaran |
21421 |
68.98000195 |
+16.50976164 |
+64.7 |
-187.2 |
0.867 |
| Fomalhaut |
113368 |
344.41177323 |
-29.62183701 |
+333.2 |
-163.2 |
1.166 |
| Antares |
80763 |
247.35194804 |
-26.43194608 |
-9.3 |
-21.8 |
1.02 |
Positional
data from Hip.Cat. (J1991.25) [4], proper motion data from PMFS [5], and
magnitude data from HIC2 [6].
About
the rigorous computation of the secular motion of a star, and the
conversion of coordinates for long-term intervals, you may find a
tabulation of formulas on a separate
page, with examples. What is given
there is the most accurate method, which complies with the recent
theories. It will be wise to follow the rigorous computation instead of
approximations, to obtain reliable results. Most of the simulation
programs fail to compute the positions with full precision. Therefore, I
should warn the reader not to rely on such programs.
Four
Stars and the Angular Distances
The
ideal configuration, which is expected to verify the hypothesis, is to be
an exact rectangular cross-shaped figure on a certain plane, where the
four stars are placed on the corners of the cross. But, since the
possibility of such an exact configuration is very low, we will first
examine the angular distances between these stars to understand the
pattern of the configuration.
To
demonstrate a contemporaneous sample of the angular distances (u) between
these stars, we may apply this spherical equation with the coordinates at
the epoch J1991.25, as given above:
cos
u = sin d1 sin d2
+ cos d1 cos d2
cos ( a1 - a2
)
With
±0.5" error, results are:
| ANTARES |
99°
56' 17" |
REGULUS |
| 82°
51' 26" |
Fom
158° 57' 32" Reg |
80°
07' 54" |
| Ant
169° 57' 48" Ald |
| FOMALHAUT |
93°
31' 47" |
ALDEBARAN |
As
it is seen above, the positions of the stars are not shaping an exact
cross. Sum of all six angles is 685° 22' 43", which is 34°
37' 17" (4.81 %) less than the ideal sum 2 × 360°. The limit
for such combinations is 6° per angle, and total limit for six angles is
36° (5 %). Then, we may assume that the combination is hardly within the acceptable
limits.
Triple
combinations of four stars give these results:
|
Ald
- Reg - Ant |
350°
01' 58" |
-
9°
58' 02" |
2.77
% |
passed |
| Ald
- Reg - Fom |
332°
37' 13" |
-
27°
22' 47" |
7.61
% |
failed |
| Ald
- Ant - Fom |
346°
21' 00" |
-
13°
39' 00" |
3.79
% |
passed |
| Reg
- Ant - Fom |
341°
45' 15" |
-
18°
14' 45" |
5.07
% |
failed |
| Total |
1370°
45' 26" |
-
69° 14' 34" |
4.81
% |
passed |
Binary
combinations of four stars give these results:
| Ald
- Reg |
80°
07' 54" |
-
9°
52' 06" |
10.96
% |
failed |
| Ald
- Ant |
169°
57' 48" |
-
10°
02' 12" |
5.58
% |
failed |
| Ald
- Fom |
93°
31' 47" |
+
3°
31' 47" |
3.92
% |
passed |
| Reg
- Ant |
99°
56' 17" |
+
9°
56' 17" |
11.04
% |
failed |
| Reg
- Fom |
158°
57' 32" |
-
21° 02' 28" |
11.69
% |
failed |
| Ant
- Fom |
82°
51' 26" |
-
7° 08' 34" |
7.94
% |
failed |
| Total |
685°
22' 43" |
- 34°
37' 17" |
4.81
% |
passed |
Binary
list is indicating that irregularity caused mostly by the position of
Regulus (32.9 %), and of Antares (24.0 %), and Fomalhaut (23.0 %), and
Aldebaran (20.0 %). But, triple list indicators are showing that
the most unbalanced position has the star Fomalhaut (28.5 %), and
then Regulus (26.8 %), and Aldebaran (24.5 %), and Antares (20.1
%). When both combinations are merged, irregularity percentages
become like this: Regulus (29.9 %), Fomalhaut (25.8 %), Aldebaran
(22.3 %), and Antares (22.1 %).
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Four Stars and the Ecliptic Plane
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Due
to the apparent paths of solar system objects, ecliptic plane becomes
more preferable than the equatorial plane. In that case, we may
reconsider the positions of the four cardinal stars by projecting them
on the ecliptic. (You may find the formulae to convert equatorial
parameters to ecliptic reference system on a separate
page.) Applying the figures given in the Hipparcos Catalogue as
above, we obtain these values for the ecliptic and the epoch of the date
J1991.25 (which is April 02, 1991 13:30 UT, if you wish to check the
reliability of commercial astrology programs):
| Name |
l
(arcdeg) |
b
(arcdeg) |
Vm |
Grade |
| Regulus |
149.70747148 |
+0.46456057 |
1.360 |
1.360 |
| Aldebaran |
69.66684753 |
-5.46794245 |
0.867 |
0.890 |
| Fomalhaut |
333.73701592 |
-21.13451720 |
1.166 |
1.502 |
| Antares |
249.64011173 |
-4.56879252 |
1.020 |
1.036 |
Rightmost
column figures named grade are based on the formula G = Vm
+ 5 ( 1 - cos b ) to estimate the relative
value of the star in respect to ecliptic plane, which is introduced by
me for the first time in 1992, to evaluate the importance of the
stars in astrological applications. The limiting constant 5 is to assign
a grade value of 5 to a star with magnitude 0.0 and latitude ±90°, when
a similar one with latitude 0° bears the grade value 0. The scale for
this evaluation is reverse, i.e. the importance
will be lessened as the grade increases. In practical usage, stars with
G > 5.0 has no astrological importance. A similar evaluation is to be
applied to the individual charts by substituting the ecliptic latitude (b)
with the elevation (h) of the star, as well as of any celestial object,
which I may explain in another article, some time in the future.
Here
are the angular distances based on ecliptic projections, namely the
separations between ecliptic longitudes of the stars, disregarding
the latitudes (with
±0.5" error):
| ANTARES |
99°
55' 58" |
REGULUS |
| 84°
05' 49" |
Fom
175° 58' 14" Reg |
80°
02' 26" |
| Ant
179° 58' 24" Ald |
| FOMALHAUT |
95°
55' 47" |
ALDEBARAN |
According
to these figures, sum of all six angles is 715° 56' 37", which is
4° 03' 23" (0.56 %) less than the ideal sum 720°. Now, we may assume that the combination is
safe within the acceptable
limits.
Triple
combinations of four stars with projected angles give these results:
|
Ald
- Reg - Ant |
359°
56' 48" |
-
0° 03' 12" |
0.01
% |
passed |
| Ald
- Reg - Fom |
351°
56' 27" |
-
8° 03' 33" |
2.24
% |
passed |
| Ald
- Ant - Fom |
360°
00' 00" |
0°
00' 00" |
0.00
% |
passed |
| Reg
- Ant - Fom |
360°
00' 00" |
0°
00' 00" |
0.00
% |
passed |
| Total |
1431°
53' 15" |
-
8° 06' 45" |
0.56
% |
passed |
Binary
combinations of four stars with projected angles give these results:
| Ald
- Reg |
80°
02' 26" |
-
9° 57' 34" |
11.07
% |
failed |
| Ald
- Ant |
179°
58' 24" |
-
0° 01' 36" |
0.01
% |
passed |
| Ald
- Fom |
95°
55' 47" |
+
5° 55' 47" |
6.59
% |
passed |
| Reg
- Ant |
99°
55' 58" |
+
9° 55' 58" |
11.04
% |
failed |
| Reg
- Fom |
175°
58' 14" |
-
4° 01' 46" |
2.24
% |
passed |
| Ant
- Fom |
84°
05' 49" |
-
5° 54' 11" |
6.56
% |
passed |
| Total |
715°
56' 37" |
-
4° 03' 23" |
0.56
% |
passed |
Binary
list is indicating that irregularity caused mostly by the position of
Regulus (32.5 %), and of Aldebaran (23.6 %), and Antares (23.5 %), and
Fomalhaut (20.5 %). But, triple list indicators are showing that
the most unbalanced position has the star Regulus (33.3 %), Aldebaran
(33.3 %), and Fomalhaut (33.1 %), and then Antares (0.2
%). When both combinations are merged, irregularity percentages
become like this: Regulus (32.9 %), Aldebaran (28.4 %), Fomalhaut (26.8
%), and Antares (11.8 %).
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Conclusion of Part One
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In
our day, it seems hardly possible to regard these four stars as the
cardinal ones, which are supposed to hold the four corners of the heaven.
Because, neither the true angular distances nor the separations on the
ecliptic plane are verifying this assumption. Especially, the position of
Regulus is quite astray than the expected point. The next one, which does
not fit to the combination is Fomalhaut. Here we have a shifted
Regulus-Fomalhaut axis, preventing us to believe in the cardinality of
these stars. It seems that there is a trace of dispersion with the present
fringe of the combination.
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But,
before jumping into conclusions, it may be wise to consider the wanted
configuration some time out of our era. Because, during long intervals of
time, the positions of the stars are changing, due to secular movements.
And the components of the shifted axis are the most fast moving objects
among these four.
If we just take a glance at the total proper motions
of these stars, the sequence may give some idea to us: Fomalhaut
371", Regulus 249", Aldebaran 198", and Antares 24" in
one thousand years. When we compare the angular distances with the
directions of these motions, a rough estimation gives a clue about the
possibility of a more fitting configuration, not today but some 70,000
years ago!
On
the left is the schematic representation of the present positions of four
stars, projected on the equatorial plane. Angles are showing the
present deviations from quadrants based on the position of Aldebaran, and
the arrows are showing the direction of the movement of the stars when we
go back in time, with angular speeds reduced to the equatorial plane, in
terms of right ascension as arcseconds per thousand years.
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Second
part of this article will let you to examine the existence of such a
possibility, in numbers and graphics.
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References:
[1]
Bretagnon, P. et al.: 2003, "Expressions for Precession Consistent with the
IAU 2000A Model". A&A 400,785
[2]
"Star-Names and Their Meanings" by R. H. Allen - Stechert -
1899
[3]
"Die Anfänge der Astronomie" by B.L. Van der Waerden - Groningen -
1966
[4]
I/239 The Hipparcos & Tycho Catalogues - ESA, 1997.
[5]
I/266 Proper Motions of Fundamental Stars - Gontcharov, 2001.
[6]
I/196 Hipparcos Input Catalogue, 2nd version - Turon, 1993.
Copyright © 2004-2008 Haluk Akcam. All rights reserved.
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