An introduction on the movements of the moon

By Thomas Schmidt

The phases coinciding with high northern declinations are first quarter in spring, new moon in summer, last quarter in autumn, and full moon in winter.

The below picture shows a magnified view of the declination range from 28° to 29° for the same time interval. The blue line represents the envelope of declination of the northernmost point of the moon's orbit (the moon itself will follows the red, green, black and grey curves of course), assuming that the inclination is constant and the nodes regress with constant speed. Under these conditions, the envelope of the orbit would reach its extreme declination when the ascending node aligns with the vernal equinox (that is, the node's ecliptical longitude passes through zero, see top diagram) and the lunar inclination therefore simply adds to the obliquity of the ecliptic (23.44° + 5.15° = 28.59°).

The blue line reaches a maximum every 18.61 years, the duration of one complete nodal cycle. This is also the mean duration of one standstill limit cycle, but the details are somewhat more complicated than this simple picture.

In reality, a perturbation term with amplitude 8.65' (plus other, minor terms) and a mean period of ca. 173 days is superimposed on the inclination, and the speed of the nodes shows small wobbles, too. As the result the envelope of the northernmost points of the lunar orbit (gray line) oscillates around the idealized blue line, and the real northern extreme of the lunar orbit is reached close to the maximum of one of these wiggles. The wobble maxima occur when the sun passes through one of the lunar nodes (see top diagram), hence the above mentioned period of ca. 173 days. As in the vicinity of a major standstill limit the ascending lunar node is close to the vernal equinox, this means that the sun is close to an equinox, and thus major standstills are biased to occur close to the equinoxes, despite their mean period of 18.61 years which is not an integer multiple of the distance between two equinoxes. In effect, the standstill limit cycle is found to be of a length clustered around 18, 18,5 and 19 years.

The declination of the moon oscillates monthly between the extrema allowed by the current disposition of the orbit (the above mentioned envelope) and is thus constrained by the variable circumstances discussed so far.
However, the inclination of the orbit changes appreciably on time scales comparable to a month, and therefore a high inclination of the orbit may happen while the moon is far from the northernmost point of the orbit, and it may already be gone when the moon reaches this point. So the question of the moon reaching extreme declinations is a matter of successes and missed opportunities: the inclination of the orbit must be high and the moon must pass through the northernmost point of the orbit.

In 2006 it happens that two perturbation wobbles lie roughly symmetrical to the idealized unperturbed declination maximum  and are therefore of comparable height. The below picture shows these two wobbles in detail. The moon misses the maximum of the March wobble by about half a month, but in the September wobble it happens to pass through the northernmost point of its orbit close to the time where the orbit has maximum inclination.
The maximum northern geocentric declination of the moon for the 2006 standstill season is therefore reached on September 15^th, 1:28 TD (28°.7227), with the maximum of April 4^th, 7:38 TD (28°.7150) as a close runner-up.

However, a megalithic astronomer who is (supposedly) constrained to measure the position of the moon at the instants of rising or setting azimuth will in general miss these maxima declinations and can only assess the maximum azimuth of those positions of the moon that are observed at rising or setting, which is a relatively small and very specific set of observations. The below picture displays the rising declinations of the moon computed for Callanish I and the 2006 standstill season as gray squares. It is immediately evident that the maximum observed rising declination is for most months markedly less than the maximum declination reached in that month (but at a time where the moon didn't happen to rise).

On average, the moon covers 13° along its orbit per day, so from one moonrise to the next the moon will have wandered about 13° (very roughly). Imagine that the orbit is subdivided into these 13°-segments, and that moonrise happens whenever the moon passes from one segment to the next. The northernmost point of the orbit will lie somewhere in one of the segments. It may happen to lie right at a segment boundary, then the corresponding moonrise will be an extreme one, and the declination of the rising moon will be the maximum declination allowed by the current orbit. But the northern-most point may also lie in the middle of a segment; then the passage of the moon through this month's maximum declination happens between two risings and goes unseen by the megalithic astronomer. At the preceding or the following moonrise, the moon will be 6.5° away from the northernmost point, and its declination will be 0.2° less than the maximum. For all other positions of the northern-most point in a segment, one of the boundaries is closer than 6.5°, so that is the maximum by which the passage through the maximum can be missed.

These cases can be identified in the below picture: the moonrises of May 1^st (5:23 UT) and May 2^nd (6:21 UT) are almost equidistant from the passage through the maximum declination on May 1^st (16:33 UT), so at both instants the moon's declination is roughly 0,2° short of the maximum (28.44° and 28.35° versus 28.61°).

On the other hand, on April 4^th, May 29^th, August 18^th and September 15^th the moon rises when it is close to maximum declination, and the observer will note extreme rising azimuths. Assuming that the observer is not hindered by bad weather or other obstacles, his observation report will finally comprise all the gray squares in the diagram, and the northernmost moonrise in this set of observations will be the one of April 4^th, 2006. This moonrise happens to almost coincide with the maximum declination the moon reaches in the March/April perturbation wobble, the time difference is only five minutes.

As discussed earlier, the moon reaches an even higher declination in the September wiggle, on September 15^th. But here the closest moonrise happens 5 hours earlier than maximum declination, and the moonrise declination is therefore 0.04° less than the maximum.

So although the moon reaches a higher declination on September 15^th than on April 4^th, the moonrise on April 4^th will have a higher declination than the moonrise on September 14^th.

For convenience and consistency, the foregoing discussion was considering the behavior of extreme geocentric declination. The analysis of megalithic structures however involves extreme rising or setting azimuths which in turn depend on topocentric longitude. The change in longitude may give slightly different ranking of extreme azimuths than the ranking of geocentric declinations. The definitive analysis for each location should therefore be done in terms of azimuths.