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Lunar declination and azimuth
The lunar declination (and thus its azimuth) is determined by
many aspects:
- the ecliptic (with a constant influence on declination of
23.44°,
at
2000 CE)
- the lunar inclination (with a maximum influence on declination of
5.1454°,
the maximum is independent on the epoch, the actual value is depending
on the nodal cycle)
- the perturbation in the lunar inclination (depending on the
relative
position
of sun and moon, with a maximum of 10').
- the parallactic fluctuation (amounting to 6' over a 180 Year
period:
Ruggles
[1999]).
- others (hopefully these have a smaller contribution than 1')
Remember that the error in the moon's position around 4000 BCE is around
13', so influences smaller than 1' are not important.
The relative effects of the first three can be calculated
independently,
but the overall effect is a combination of:
- the positions of moon and sun (determining the perturbation),
- the position of the lunar nodes with
respect to
the
solar vernal/autumnal points and
- lunar parallax
Perturbation of the lunar inclination
Introduction
This section only talks about the perturbation of the lunar inclination
(which has a micro effect on the actual moon declination), remember
that
the moon's path is also determined by it normal changing inclination.
Theory
A formula for the perturbation of the lunar inclination is given by
L.V.
Morrison ([1980] and pers. comm.
[2001]):
Perturbation = 8.65cos(2Lsl) - 0.70cos(2(Lml-L
sl))+ 0.65cos(2Lml) [']
Lsl: Sun's longitude measured from the longitude of the
lunar
node
Lml: Moon's longitude measured from the longitude of the
lunar node
Remark: North [1996,
page 563] made a mistake by talking about the
ecliptic longitude of sun and moon. Happely this error does not
really
effect the rest of the points North makes.
Remark: A. Thom and
A.S. Thom [1978]
copied an erroneous formula from M. A. Danby [1962] (the second
argument
had a + instead of a -), which changes the big wobble values to resp.
+10'
and -8.7' instead of +8.7'
and -10'. Due to this Thom used an inclination
between 5° 18' 43"
and 5° 0' 1", while it should have been: 5°17' 25" and 4°
58'
43". If this causes problems to Thom's high precision deductions, I
don't
know.
If we graph this formula, we get the following:
(the graph repeats itself after 180° solar longitude and after
180° lunar longitude)
Important solar and lunar events:
- First and last quarter moon
90° or 270° difference between solar and lunar longitude - New
moon
0° difference between solar and lunar longitude - Full moon
180° difference between solar and lunar longitude - Possibility
of eclipses
If the Sun is inline with the lunar nodes (Ls1 of 0° or
180°), which is at the maximum of the perturbation, and if new or
full Moon there is a possibility of an eclipse
Different variations can be noticed in the moon's
perturbation:
- Any lunar phase around the major/minor standstill limits and any
location
of the sun.
A variation in perturbation of some 19': from +8.7' to -10' (big wobble)
- Lunar phases around moment that the sun is near lunar nodes
The variations in perturbation are so small (around 0.1') that they are
not noticeable without modern tools.
Remember that a change in air pressure of 8 mbar already provides the
same change.
- Lunar
phases around moment that the sun is 90° from the
lunar nodes
The variations in perturbation are so small (around 2.6') that they are
not noticeable without modern tools.
Remark: This is different from what North [1996,
page 564] is telling in his text, but not in his pictures.
Remark: A. Thom
[1973]
copied an erroneous formula from M. A. Danby [1962] (the second
argument
had a + instead of a -), which might have shown him that full/new Moons
would have the maximum declination (the pink line would have been
around +10'). As can be seen, the quite stable blue line is on top (at
around +8.7'), but slightly peaking at quarter-ish moons.
Conclusion on perturbation
One can use all the moon phases around a standstill limit, because
there
are no decernable differences between the inclination at any of the
moon
phases. The largest perturbation differance is seen between the sun
near
a lunar node and the sun at 90° from a lunar node (with a period of
~ 0.5 ecliptic year ~173.31 Days and maxima at quarter lunar phases and
minima at full/new lunar phases).
Standstill limits due to the lunar nodal
cycle
The major standstill limit of the moon can be reached if the lunar node
is near the vernal (or autumnal) point, and thus the Moon has its max.
distance
from the equator, equal to a declination at present days of 23.44°
+ 5.1454°= 28.59°.
The minor standstill limit of the moon can be reached if the lunar
node is near the vernal (or autumnal) point, and thus the Moon has its
min.
distance from the equator, equal to a declination at present days of
23.44°-
5.1454° = 18.29°.
Declination of moon
When combining the effects of the lunar nodal cycle and the
perturbations
we get the following major and minor standstills limits:
- major standstill limit
This will happen at the moment the moon is near a quarter moon and
the lunar node is near the vernal (or autumnal) point. The moon is at
his
highest point in its orbit and combining this with lunar phase, the sun
is near equinox.
The (geocentric) declinations of the major standstill limits at present
days are
28.59° + 0.145° = 28.74° (or -28.74°) - minor
standstill limit
This will happen at the moment the moon is near a full/new moon and
the lunar node is near the vernal (or autumnal) point. The moon is at
his
lowest point in its orbit and combining this with lunar phase, the sun
is near solstice.
The (geocentric) declinations of the minor standstill limits at present
days are
18.29°- 0.145° = 18.14° (or -18.14°)
Lunar parallax
The lunar parallax comes into play, because we are observing the moon
rises
and sets (see below). The mean horizontal parallax changes the apparent
altitude of the
moon, because we are not observing the moon from the center of the
earth
(max: 60'.24 and min. 53'.97). This topocentric altitude makes that the
moon appears lower in the sky and thus the rise and set azimuths of the
moon will be more southern.
The most important though for this discussion is that the parallax
is also dependent on the distance between earth and moon, so the lunar
parallax and thus the moon set and rise azimuths are also dependent on
the anomalistic month.
Azimuth of moon
Calculating the resulting azimuth of the moon as a function of the days
after a major standstill limit (one can see here only the maximum
reachable
values of the moon's azimuth and not the actual values):
There is a difference in the
major/minor
standstill limits in declination (which is the definition of major
standstill
limit) and the method I am using; which is the major/minor azimuth
standstill
limit. This is because the rise/set moments of the moon do not
necessarily
have to be the same moments as reaching the extreme declination as seen
in the below picture. Also different locations may experience different
dates for the major/minor azimuth standstill limits.
The grey squares are the lunar set and rise events.
The above picture has been made by Thomas Schmidt
These can differ by a few periods of the tropical month!
On my web-site I use the following convention: When I talk about
major standstill limit it is always in the standard definition, thus
looking
at declination, and if I use 'major/minor azimuth standstill', I am
looking
at the extremes in azimuth.
Using the azimuth is perhaps more relevant for this web site, because
I think that that azimuth value could have been observable by neolithic
man and not the (modern grid of the) declination.
The following can be deducted from all the above:
- The big wobble is due to the 9' declination wobble of the
perturbation
of the moon's inclination. The maximum of this wobble is at a
first/third
quarter moon and the minimum is at a new/full moon. The period is ~ 0.5
ecliptic year or ~173.31 Days.
I have the opinion that a difference of some 0.9° is measurable
with a megalithic building. According to Thom [1973]
megalithic people did, according to Ruggles [1999]
this can't be proven (yet?). - Megalithic buildings were perhaps
able to registrate this
observed difference
(0.9°) between moon's azimuth or a way to predict
eclipses (like said by Thom [1973, page 19])
- The small wobbles (0.1' and 2.6') are due to the changing
positions of
moon and sun (resulting in different phases of the moon). The small
wobbles
are too small to be measured by means of megalithic buildings.
- If the big wobble would not be there, the
experienced
periodicy of the minor/major azimuth standstill limits would be ~18.61
Years (lunar nodal cycle). Due to the big wobble, the experienced
periodicy
of the minor/major azimuth standstill limits is clustered around ~18,
~18.5
and ~19 years (due to the periodicy of the wobble of half an eclipse
year).
If one only looks at rises/sets of the moon during day time or night
time, it changes to a ~18 and ~19 year periodicy.
In the below picture this can be seen for the moon rises calculated
using JPL and gathered by Jo
Coffey over a 180 year period. Dates are Julian dates (that is why
they look out of date with equinox, but they are not).
While checking this period for major
azimuth standstill limits and
the
following periods can be seen (green for a
time differance very close to 19 tropical years [Metonic cycle] and purple
for a time differance close to 18.03 tropical year [Saros cycle]):
- Day and night rising times:
18.5 - 19 - 18.5 - 19
- 18.5 - 18 - 19
- 18.5 - 18.5 - Day rising times:
18 - 19 - 18-
19
- 19 - 18 -
19
- 19 - 18 - Night
rising times (most likely
witnessed):
19 - 18 - 19 - 19 - 18 - 19 - 18 - 19
- 18
The distribution of Saros and Metonic cycles within the series is quiet
interesting!
Because the set and rise azimuth of the moon is dependant on the
tropical
year, nodal cycle, synodical, anomalistic and draconic month, it is
expected
that the total azimuth shows Metonic
cycles
(depending on tropical years and synodical months) and Saros
cycles (depending on synodical, anomalistic and draconic months).
- It is perhaps interesting to see that such a sequence of periods
was
also proposed by Hawkins to explain his 'computer' at Stonehenge (see
e.g Brown
[1978], page 125). I wonder if the
sequence Hawkins has proposed, has
ever been related to this manifestation of the lunar nodal cycle?
- Perhaps
this 18 - (18.5 -) 19 year wobble in time has been more
easily
observed, than the 0.9° wobble in azimuth?
- The question arises if an accurate
determination of the real
major/minor
azimuth standstill limit date within the 18 - (18.5 -) 19 year
wobble
can be determined using a passage cairn or free standing stones.
For determining the real date of azimuth standstill limit, there is
a window of some 0.4° azimuth in which the minor/major azimuth
standstill
limit for that 18.61 Year period will occur (between 147° and
147.4°,
see above picture).
But the accuracy is not that big, because the boundaries between max.
light intensity of the moon light patch and real shadow is quiet faint.
This is also due to size of quarter moon, also some 0.25°, which is
in the same order as the difference we want to measure. With a 12
m passage this is some 5 cm change in shadow on a backwall (if you use
the passage walls, this 5 cm will of course grow
to bigger figures, so the faint difference between moon light and
darkness
will make it less accurate). Remember that the standstill limit does
not
have to be at the same moment as set or rise, this will also introdue a
small error (around 0.1°). The variation of altitude due to
astronomical
refraction (0.07°) and parallax (0.1°) could also make things
less
accuracte (this results in a variation of azimuth of 4 times bigger).
So determining the real date and thus the periodicy (of 18,
18.5 and 19 years) of major/minor azimuth standstill limits will not be
that simple using shadows on a passage (back-)wall. - A comparison of calculations
is done
of major azimuth standstill dates. This also stresses the idea that one
date for the major azimuth standstill limit is not practical. This idea
is also mentioned by Thom ([1973], page
18):
"The Moon, it is true, in no sense stands still, but for about a
year the limiting declinations do not vary more than 20 arc minutes, so
that for month after month the Moon's declination goes through almost
the
same cycle."
Acknowledgments
Thanks to the phenomenological
work of Jo Coffey (except that the actual major azimuth standstill
limit dates mentioned [in BCE and CE periods] are wrong but she is
working
on this; the form of the graphs though are oke) and the help of Leslie
Morrison and Thomas Schmidt, I was stimulated to make this theoretical
page. Another very related page is on the Hopewell
lunar astronomy (except that in this page in most instances draconic
month has to be replaced with tropical month).
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Last major content related changes: April 7, 2001