RASC Calgary Centre - Lunar Angles 101

By: Larry McNish
Page last updated: January 3, 2017
(Page originally created July 10, 2009)
(All diagrams and charts by the author.)

Questions:

Why does the Full Moon sometimes appear so low in the sky compared to other Full Moons?
Why doesn't the Moon rise and set due East and due West?
Why can't you add the tilt of the Moon's orbit to the tilt of the Earth to get the Moon's angle?

Answer:

The exact position of the Moon (as seen against the sky) is one of the most complicated astronomical mathematical calculations, since it is affected by both the Sun and the Earth (and the Moon also effects the Earth's position).

One thing that is often overlooked is that, whereas the orbit of most moons in the Solar System are aligned with their parent planet's equator, our Moon's orbit is aligned with (i.e. tilted relative to) the Earth's orbit, not its equator. The diagram below (from The Sun, The Moon, and Eclipses) shows this - the Moon's orbital plane is "constant" i.e. you do not simply add it to the tilt of the Earth's equator.

First, consider that the Moon's orbit (actually the orbital plane) is relatively stationary, like the Earth's. So, if we assume that at one point in the Earth's orbit around the Sun, the Moon's orbit is tilted "up to the left" as I show in my diagram above. Then, when the Earth is on the other side of the Sun, the Moon's orbital plane is still tilted "up to the left". However, the Earth is also tilted at a relatively constant angle independent of orbital position. So sometimes these two tilts "line up" and sometimes they are "opposite".

Diagram 1 - Winter in the Northern hemisphere

In Winter, the Northern Hemisphere is tilted "away" from the Sun.
If you were at latitude "A" on the rotating Earth, then on the night of the Full Moon you would look up (close to your perpendicular, or Zenith) to see the Full Moon (right side of the diagram).

Diagram 2 - Summer in the Northern Hemisphere

Six months later, during the Summer, the Earth is on the "other side" of the Sun and the Northern Hemisphere is tilted "towards" the Sun. If you were still at latitude "A" on the rotating Earth, then on the night of the Full Moon you would look down towards your southern horizon (i.e. much further from the perpendicular) to see the Full Moon (left side of the diagram.

As seen in the two diagrams, the position in the sky for the Full Moon varies considerably from near the southern horizon, to nearly overhead (depending on your latitude).

The graph below shows the extremes reached in March and April of 2006. Since the Moon's "altitude" in the sky depends on its orbit, the observer's location and the time of day and day of month, the graph below simplifies this by plotting the Moon's position as a function of which Earth Latitude it is directly over during the month. As can be seen in the graph, it reached a "far northern" position above Earth latitude +28.7167° and a "far southern" position above Earth latitude -28.6667° half a month later. This resulted in one of the "lowest" (nearest the horizon) moons in years, as seen from Calgary.

Similarly (though it would take many more diagrams to show it) the position of the moon at 1st Quarter, 3rd Quarter, Sunrise, Sunset etc. all vary in "maximum height" over the course of a year.

If the Moon is "low" in the sky, then it will rise south of East and set to the south of West because your horizon obscures it as it rises and sets.

If the Moon is "high" in the sky, then it will rise north of East and set to the north of West because it is much more northerly against the horizon and rising at a much steeper angle.

At intermediate dates, things can line up so that the Moon rises almost due East and sets almost due West (depending on your latitude and the timing).

So that's it - right?

Unfortunately, no. When I said above that the Moon's orbit is "relatively stationary", I over-simplified. If you think of the Moon's orbit as a dinner plate held at an angle and just touching a tabletop, then what really happens is more like when you spin the plate like a top and let it wobble on its edge on the table. Although it's turning fast, it's also wobbling so that the point that touches the table slowly rotates around in a circle.

Since the Moon is much larger than your average dinner plate, it takes a long time for the Moon's orbit to complete a wobble - 18.6 years. This is called the "precession of nodes". It's caused by the gravitational tug of the Sun trying to flatten out its orbit, and by the tug of the Earth trying to make it orbit our equator. But due to the Moon's large angular momentum neither can win - all they can do is create a net torque on the Moon's orbit which causes the whole orbit to rotate around the Earth over the 18.6 year cycle.

The net effect of this is that both the middle two diagrams above will not apply to Winter and Summer as I showed after a couple of years - the dashed line shown as the orbital plane of the Moon will rotate about the Earth's centre over this long period, and half way around it will be tilted the other way. This was actually discovered by the ancients - that the very highest and very lowest Full Moons (as well as Solar and Lunar eclipses) repeated after about 19 years and they called it the Saros Cycle. (Since the Moon's orbit rotates in this manner, it is not directly or constantly related to the Earth's perihelion or aphelion.)

So, although the angles of the Earth's tilt (23.4 degrees) and the angle of the Moon's orbit (5 degrees) are fairly small, the way things work they do not add together, and the Moon can be found almost all over the sky. The mathematical routines I use to calculate the rising and setting times of the Moon on my darksky page take all this (and more) into account, including the Earth's elliptical orbit motion and the observer's longitude and latitude and they are some of the most complicated programs I work with.

Just to show how strange this phenomena is, take a look at sunset_moonrise.htm especially the graph of the time of day of Sunset and Moonrise. You'll see that not only does the Moon's position change as I explained above, but the times of Moonrise oscillate wildly over the course of a year.

To be really precise:

The Moon does not orbit the Earth - instead, both the Earth and Moon orbit a common point called the Barycentre as described on the page howfast.htm - See section 2 and particularly the animation I created to show the Barycentre motion. Additionally the Earth orbits a Sun which is orbiting a Barycentre caused by the relative motions of the Sun and Jupiter. In general, however, the Moon is much smaller than the Earth and these barycentre distances are smaller than the respective orbits so things are "relatively" stable.

The Moon's mean inclination of orbit to ecliptic = 5.14° (ranging between 4.99° and 5.30° as per Jean Meeus, Mathematical Astronomy from which most of my calculation routines were originally based.

And, of course, the Earth itself is wobbling (precessing) over a period of 25,770 years or 1° every 71.58 years as explained at the bottom of radecl.htm . Yes, 1 full degree in less than 72 years. This means that all angles measured relative to the Earth must be updated regularly. When I started in astronomy we were using "Epoch 1950" for the coordinates for all the stars and deep-sky objects. Everyone now uses "Epoch 2000" coordinates except those trying to hit the planets - they have to use the "real" epochs corresponding to their launch and arrival dates. Due to precession, the "North Star" Polaris will soon be nearer the actual North Celestial Pole than it ever has, and after that our pole will start pointing away from it.