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Lunar declination and azimuth

The lunar declination (and thus its azimuth) is determined by many aspects: 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:

  1. the positions of moon and sun (determining the perturbation),
  2. the position of the lunar nodes with respect to the solar vernal/autumnal points and
  3. 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 [1973, 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'. Due to this Thom used an inclination between 5° 17' 25" and 5° 0' 1", while it should have been: 5°17' 25" and 4° 58' 43". If this causes problems to Thom's high precise 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 moments:

One can be see from the graph that the perturbation of the moon's inclination changes very slowly over time from +8.7' to -10' (big wobble).
When the sun is around the lunar node, the lunar perturbation changes only very slowly due to the lunar phases (some 0.1', with the maximum at quarter moons).
When the sun is 90° of the lunar node, the lunar perturbation changes slowly but a little bit more than the sun near the lunar node (some 2.6', with the minimum at new or full moons).
Remark: This is different from what North [1996, page 564] is telling in his text, but not in his pictures.

So there are a few periods that one could notice in the moon's perturbation:

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 major perturbation differance is seen between the sun is 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 with the moon at 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 with the moon at 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:

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:

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