by Richard Rosenberg
Recently you may have noticed that the Moon seemed to be unusually high or low in the sky. What causes this? When does it happen? Is there a pattern? What are the consequences, if any, of this? Can we learn anything from studying this behavior?
If we carefully watch the Moon for a month as it goes through its cycle of phases, we will detect a day when it is highest in the sky, and another (about two weeks later) when it is lowest. At its highest, the Moon is seen to rise well north of east and set a similar distance north of west. On these days the Moon stays above the horizon for a long time, well over half a day. The opposite occurs when the Moon is low — it rises and sets well south of the cardinal points and is up much less than half a day.
We also note the sequence is continuous. That is, if each night we mark the locations along the eastern horizon where the Moon rises or along the western horizon where the Moon sets, we will see a continuous progression. Between “high Moon” and “low Moon,” these points progress southward from day to day. Between low Moon and high Moon, the points progress northward. We will also discover the distance north of a high Moon is about the same as the distance south of a low Moon.
This observation may have been made by ancient inhabitants of the British Isles. Archaeologists have discovered recumbent stones seeming to point to locations on the horizon where a high or low Moon rose or set. These stones have been dated to the late third millennium or early second millennium BC!
If we continue studying the Moon the following month, there will again be a day when the Moon is highest and another about two weeks later when it is lowest. If we measure the positions of the highest and lowest Moons when they rise, we again find the high Moon well north of east and the low Moon well south by almost the same amount. But if we are careful, we can detect something curious. The distances north and south have changed slightly from those of the last month. If it has increased, this month’s high Moon will be a little higher than last month’s, and this month’s low Moon will be a little low. On the other hand, if the distance has decreased, this month’s high and low Moons will be less extreme than the previous month’s.
If we continue this study month after month, we find the trends continue — the extremes keep growing or keep getting smaller. But not forever. If the extremes have been growing, eventually they reach a maximal value, then start to decrease. On the other hand, if the extremes have been decreasing, they reach a minimal value, and then begin to increase. Eventually the trends reverse again — after a generation of moonwatching, we find a cycle which turns out to be 18.61 years from maximal value to minimal back to maximal (or from minimal to maximal to minimal).
What on Earth (or the Moon or the universe) causes this behavior? At this point, we leave the cave man behind. We can figure out the answer with a lot of geometry and a little celestial mechanics.
The Moon, like other objects in the sky, appears at its highest above the horizon (its altitude) when it reaches the meridian, the great circle passing through the zenith and the north and south points on the horizon. This is the point halfway between moonrise and moonset. We can measure how high the Moon is when it crosses the meridian (where 0° is the horizon and 90° is the zenith).
The celestial equator is the projection of Earth’s equator into the sky. It passes through the east and west points on the horizon and reaches a maximum altitude of (90 – λ)°, where λ is the observer’s latitude. For example, at New York’s latitude of about 40°, the celestial equator would be 50° above the horizon at the meridian.
Consider the appearance of the Sun in our sky. The path it follows in the course of a year is called the ecliptic. This path is also the intersection of the plane of Earth’s orbit with the sky. We are familiar with the phenomena of a “high Sun” and “low Sun,” though we don’t use that terminology. In summer, the Sun is quite high in the sky; in winter, very low. If we check the Sun’s rising and setting points during the course of a year, we find the Sun reaches a point where it rises and sets farthest north and takes its highest path through the sky. We call the day when this occurs the first day of summer, and the Sun’s position on the ecliptic at this time is called the summer solstice. Conversely, six months later the Sun is very low, and we discover the Sun reaches a maximum distance south of east at sunrise and west at sunset. This is the first day of winter, and the Sun is at the winter solstice. There are also two intermediate positions, where the Sun rises and sets at the cardinal directions. These are called the equinoxes — the vernal equinox is the Sun’s position on ecliptic where it crosses the celestial equator moving north; the autumnal equinox is its location on the intersection of ecliptic and celestial equator when the Sun is moving south.
The reason the Sun changes its north-south position at all is due to the fact that the axis of rotation of Earth is not perpendicular to the plane of Earth’s orbit around the Sun. If the axis were perpendicular, the celestial equator (formed by projecting Earth’s equator into the sky) and the ecliptic would be one and the same. The Sun would always rise due east and set due west. In reality our axis of rotation is “tilted” at 23½° and this is the distance the Sun is north of east at the summer solstice (when the North Pole of earth’s rotational axis is facing the Sun) and south of east at the winter solstice (when it’s the South Pole’s turn). As Earth orbits the Sun in the course of a year, the Northern and Southern Hemispheres of Earth alternately point towards and away from the Sun, creating our seasons and the high and low Sun. We know it’s 23½° because from New York (for example) we measure the altitude of the Sun on the first day of summer at 73½° (50° + 23½°) and on the first day of winter at 27½° (50° – 23½°). This amount (23½°) does not change from year to year.
Now consider the motion of the Moon. As we know the Moon revolves around Earth. But the Sun exerts a strong gravitational attraction on the Moon, so determining its motion is really a three-body problem. This means that the Moon’s motion is not exactly described by a simple ellipse. It turns out the plane in which the Moon moves is related to the plane of Earth’s orbit around the Sun (that is, the ecliptic) rather than the plane of Earth’s equator. But the Moon’s orbit is titled about 5° to the ecliptic.
Two non-parallel planes intersect in a straight line. Consider the Moon’s motion in its orbital plane. At some point, it will intersect the ecliptic at a point on the line of intersection. If the Moon is heading north (from Earth’s perspective) this is called the ascending node of the Moon’s orbit. The Moon will continue until it reaches a maximally north point (again from Earth’s point of view), then turn southward, and once again will intersect the ecliptic. This second point of intersection is the descending node. (The line of intersection is in fact usually called the line of nodes.)
So with respect to the ecliptic, the Moon spends around half its time north of the ecliptic (from ascending node to descending node) and half the time south of the ecliptic (from descending node to ascending node). Since the two planes intersect at an angle of 5°, the Moon can range up to 5° north or south of the ecliptic.
We still don’t know where the Moon will be in the sky — it depends on where the nodes are positioned on the ecliptic. For example, suppose the ascending node of the Moon’s orbit happens to be at the same point as the vernal equinox on the ecliptic. This happens to occur on June 19, 2006.
Figure 1 and Figure 2 show the path of the Moon at this time. For simplicity, they show the sky as seen from a location on Earth’s equator. From day to day the Moon moves from east to west (right to left in the figures). Note the positions of the celestial equator, ecliptic, and path of the Moon. In Figure 1, the vernal equinox is at the right where the ecliptic crosses the equator heading northward. Note that the Moon’s orbit is simultaneously positioned so that its ascending node with respect to the ecliptic is almost exactly at the vernal equinox. So when the Moon reaches this point in June 2006, it will not only be moving north along the ecliptic, but north with respect to the ecliptic. This is indicated in Figure 1. In fact, when the Moon is furthest north (at the center of Figure 1) it will be about 28½° north of the celestial equator, its largest possible value, since it will be 5° above the summer solstice point, which in turn is 23½° north of the equator.
In the next week (the left side of Figure 1) the Moon will head toward the autumnal equinox. Since the Moon’s descending node at this time is almost the same as that of the ecliptic, the Moon will lie almost exactly at the autumnal equinox.
Figure 2 shows the next two weeks. Now the Moon traverses the southern part of its orbit. Again note that it travels south of the ecliptic during this time, while the ecliptic itself is south of the celestial equator. At the center of Figure 2, the Moon will be 28½° south of the celestial equator, 5° below the winter solstice point.
Figure 3 and Figure 4 show the situation if the ascending node of the Moon’s orbit is at the autumnal equinox of the ecliptic. This will happen in about halfway through the 18.61-year cycle from June 2006, in October 2015. Now when the Moon’s orbit is heading north, the ecliptic is going south. The movements are 180° out of phase. A quarter of the way around, the Moon will be 5° north of the ecliptic, but the ecliptic is 23½° south of the celestial equator, resulting in an overall position of the Moon of only 18½° south, a minimally south Moon. Two weeks later the Moon will be 5° south of the ecliptic, but the ecliptic will be 23½° north of the celestial equator, so the Moon will be 18½° north, a minimally north Moon.
If the ascending node is located on other positions of the ecliptic we will have intermediate values between 28½° and 18½°. For example, Figure 5 and Figure 6 show when the ascending node of the Moon’s orbit meets the ecliptic at the winter solstice, as in February 2011 (a quarter of the 18.61-year cycle from June 2006). Note now the maximal values are the same as those of the ecliptic (23½°) and the path of the Moon looks just like the ecliptic displaced westward (to the right). If the ascending node of the Moon’s orbit met the ecliptic at the summer solstice, the maximal values would also be 23½° but the path of the Moon would look like the ecliptic moved eastward (to the left).
The next question — why do we see all these different values? That is, why don’t we always see a maximal value of 28½° or a minimal value of 18½° or any one value in between? Why the 18.61 year cycle? From what was shown above a changing value must mean the ascending node is intersecting the ecliptic at different points, or equivalently that the line of nodes is moving. This means the Moon’s orbit is precessing, swiveling in space, making a complete revolution in 18.61 years. (Precession is another result of two bodies pulling on the Moon. Note the similarity to Earth’s precession, although the latter takes 26,000 years. In Earth’s case the effect is to change the line of nodes between the ecliptic and the celestial equator.)
Because the nodes turn out to be moving backwards with respect to the Moon’s motion, it takes only 27.21221 days for the Moon to travel from ascending node to the next ascending node. This period is called the Draconic month, and is one of the factors taken into account for the saros, a period of just over 18 years when similar eclipses repeat.
Since an eclipse can take place only if the Sun and Moon are at the same point (solar eclipse) or directly opposite (lunar eclipse), the Moon must lie on or very close to the ecliptic at an eclipse. That is, it must be near the ascending or descending node. An eclipse season is the period of time when the Moon is close enough to a node for an eclipse to occur. In 18.61 years the Moon precesses a full circle of 360°, so the precession amounts to 360 / 18.61 = 19.3° along the ecliptic = 19.0 days per year. The eclipse seasons drift backwards 19 days each year (making it possible to have three eclipse seasons in a calendar year).
If we consider a two-dimensional map of the sky with the vertical axis declination (corresponding to latitude) and the horizontal axis right ascension (longitude), the celestial equator appears as a horizontal line at 0° declination. The ecliptic, a plane making an angle of 23½° to the celestial equator, will appear as a sine curve with amplitude of 23½°. It intersects the celestial equator at the vernal and autumnal equinox points, is farthest north at the summer solstice and farthest south at the winter solstice, as shown in the figures. This curve satisfies the equation:
where δ is the declination and θ = 0° at 0 hours right ascension, 90° at 6 hours, 180° at 12 hours and 270° at 18 hours (corresponding to the right ascension of the vernal equinox, summer solstice, autumnal equinox and winter solstice, respectively).
Now the plane of the Moon’s orbit makes an angle of 5° with the ecliptic, so it will look like a sine curve with amplitude of 5° SUPERIMPOSED ON THE CURVE OF THE ECLIPTIC. We can fix this curve by specifying the phase angle φ between the ascending node of the Moon’s orbit and the vernal equinox. If the Moon’s ascending node is located at the vernal equinox (Figures 1 and 2), φ = 0°, at the summer solstice 90°, at the autumnal equinox (Figures 3 and 4) 180° and at the winter solstice (Figures 5 and 6) 270°. The equation of the Moon’s path can then be given by
sin(x + y) = sin x cos y + cos x sin y and sin 270° = -1, cos 270° = 0,
δ = 23.5 sin θ + 5 sin(θ + 270°) = 23.5 sin θ + 5 (sin θ cos 270° + cos θ sin 270°)
δ = 23.5 sin θ – 5 cos θ
To see more charts of the Moon’s positions in 2006 (when the phase angle is close to 0°) check out the centerfold of Astronomy magazine.